# Metamath Proof Explorer

## Theorem catpropd

Description: Two structures with the same base, hom-sets and composition operation are either both categories or neither. (Contributed by Mario Carneiro, 5-Jan-2017)

Ref Expression
Hypotheses catpropd.1 ${⊢}{\phi }\to {\mathrm{Hom}}_{𝑓}\left({C}\right)={\mathrm{Hom}}_{𝑓}\left({D}\right)$
catpropd.2 ${⊢}{\phi }\to {\mathrm{comp}}_{𝑓}\left({C}\right)={\mathrm{comp}}_{𝑓}\left({D}\right)$
catpropd.3 ${⊢}{\phi }\to {C}\in {V}$
catpropd.4 ${⊢}{\phi }\to {D}\in {W}$
Assertion catpropd ${⊢}{\phi }\to \left({C}\in \mathrm{Cat}↔{D}\in \mathrm{Cat}\right)$

### Proof

Step Hyp Ref Expression
1 catpropd.1 ${⊢}{\phi }\to {\mathrm{Hom}}_{𝑓}\left({C}\right)={\mathrm{Hom}}_{𝑓}\left({D}\right)$
2 catpropd.2 ${⊢}{\phi }\to {\mathrm{comp}}_{𝑓}\left({C}\right)={\mathrm{comp}}_{𝑓}\left({D}\right)$
3 catpropd.3 ${⊢}{\phi }\to {C}\in {V}$
4 catpropd.4 ${⊢}{\phi }\to {D}\in {W}$
5 simpl ${⊢}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\to {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)$
6 5 2ralimi ${⊢}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\to \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)$
7 6 2ralimi ${⊢}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\to \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)$
8 7 adantl ${⊢}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\to \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)$
9 8 ralimi ${⊢}\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\to \forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)$
10 9 a1i ${⊢}{\phi }\to \left(\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\to \forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)$
11 simpl ${⊢}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\to {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)$
12 11 2ralimi ${⊢}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\to \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)$
13 12 2ralimi ${⊢}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\to \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)$
14 13 adantl ${⊢}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\to \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)$
15 14 ralimi ${⊢}\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\to \forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)$
16 15 a1i ${⊢}{\phi }\to \left(\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\to \forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)$
17 nfra1 ${⊢}Ⅎ{y}\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)$
18 nfv ${⊢}Ⅎ{x}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)$
19 nfra1 ${⊢}Ⅎ{z}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)$
20 nfv ${⊢}Ⅎ{y}\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)$
21 nfra1 ${⊢}Ⅎ{g}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)$
22 nfv ${⊢}Ⅎ{f}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)$
23 oveq1 ${⊢}{g}={h}\to {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}={h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}$
24 23 eleq1d ${⊢}{g}={h}\to \left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)↔{h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)$
25 24 cbvralvw ${⊢}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)↔\forall {h}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)$
26 oveq2 ${⊢}{f}={g}\to {h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}={h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){g}$
27 26 eleq1d ${⊢}{f}={g}\to \left({h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)↔{h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)$
28 27 ralbidv ${⊢}{f}={g}\to \left(\forall {h}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)↔\forall {h}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)$
29 25 28 syl5bb ${⊢}{f}={g}\to \left(\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)↔\forall {h}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)$
30 21 22 29 cbvralw ${⊢}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)↔\forall {g}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)$
31 oveq2 ${⊢}{z}={w}\to {y}\mathrm{Hom}\left({C}\right){z}={y}\mathrm{Hom}\left({C}\right){w}$
32 oveq2 ${⊢}{z}={w}\to ⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}=⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}$
33 32 oveqd ${⊢}{z}={w}\to {h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){g}={h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){g}$
34 oveq2 ${⊢}{z}={w}\to {x}\mathrm{Hom}\left({C}\right){z}={x}\mathrm{Hom}\left({C}\right){w}$
35 33 34 eleq12d ${⊢}{z}={w}\to \left({h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)↔{h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)\right)$
36 31 35 raleqbidv ${⊢}{z}={w}\to \left(\forall {h}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)↔\forall {h}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)\right)$
37 36 ralbidv ${⊢}{z}={w}\to \left(\forall {g}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)↔\forall {g}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)\right)$
38 30 37 syl5bb ${⊢}{z}={w}\to \left(\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)↔\forall {g}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)\right)$
39 38 cbvralvw ${⊢}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)↔\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)$
40 oveq2 ${⊢}{y}={z}\to {x}\mathrm{Hom}\left({C}\right){y}={x}\mathrm{Hom}\left({C}\right){z}$
41 oveq1 ${⊢}{y}={z}\to {y}\mathrm{Hom}\left({C}\right){w}={z}\mathrm{Hom}\left({C}\right){w}$
42 opeq2 ${⊢}{y}={z}\to ⟨{x},{y}⟩=⟨{x},{z}⟩$
43 42 oveq1d ${⊢}{y}={z}\to ⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}=⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}$
44 43 oveqd ${⊢}{y}={z}\to {h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){g}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}$
45 44 eleq1d ${⊢}{y}={z}\to \left({h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)↔{h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)\right)$
46 41 45 raleqbidv ${⊢}{y}={z}\to \left(\forall {h}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)↔\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)\right)$
47 40 46 raleqbidv ${⊢}{y}={z}\to \left(\forall {g}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)↔\forall {g}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)\right)$
48 47 ralbidv ${⊢}{y}={z}\to \left(\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)↔\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)\right)$
49 39 48 syl5bb ${⊢}{y}={z}\to \left(\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)↔\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)\right)$
50 19 20 49 cbvralw ${⊢}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)↔\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)$
51 oveq1 ${⊢}{x}={y}\to {x}\mathrm{Hom}\left({C}\right){z}={y}\mathrm{Hom}\left({C}\right){z}$
52 opeq1 ${⊢}{x}={y}\to ⟨{x},{z}⟩=⟨{y},{z}⟩$
53 52 oveq1d ${⊢}{x}={y}\to ⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}=⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}$
54 53 oveqd ${⊢}{x}={y}\to {h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}={h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}$
55 oveq1 ${⊢}{x}={y}\to {x}\mathrm{Hom}\left({C}\right){w}={y}\mathrm{Hom}\left({C}\right){w}$
56 54 55 eleq12d ${⊢}{x}={y}\to \left({h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)↔{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)$
57 56 ralbidv ${⊢}{x}={y}\to \left(\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)↔\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)$
58 51 57 raleqbidv ${⊢}{x}={y}\to \left(\forall {g}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)↔\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)$
59 58 ralbidv ${⊢}{x}={y}\to \left(\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)↔\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)$
60 ralcom ${⊢}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)↔\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)$
61 59 60 syl6bb ${⊢}{x}={y}\to \left(\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)↔\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)$
62 61 ralbidv ${⊢}{x}={y}\to \left(\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({x}\mathrm{Hom}\left({C}\right){w}\right)↔\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)$
63 50 62 syl5bb ${⊢}{x}={y}\to \left(\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)↔\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)$
64 17 18 63 cbvralw ${⊢}\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)↔\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)$
65 64 biimpi ${⊢}\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\to \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)$
66 65 ancri ${⊢}\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\to \left(\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge \forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)$
67 r19.26 ${⊢}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge \forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)↔\left(\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)$
68 r19.26 ${⊢}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)↔\left(\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge \forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)$
69 r19.26 ${⊢}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left(\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)↔\left(\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge \forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)$
70 eqid ${⊢}{\mathrm{Base}}_{{C}}={\mathrm{Base}}_{{C}}$
71 eqid ${⊢}\mathrm{Hom}\left({C}\right)=\mathrm{Hom}\left({C}\right)$
72 eqid ${⊢}\mathrm{comp}\left({C}\right)=\mathrm{comp}\left({C}\right)$
73 eqid ${⊢}\mathrm{comp}\left({D}\right)=\mathrm{comp}\left({D}\right)$
74 1 adantr ${⊢}\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\to {\mathrm{Hom}}_{𝑓}\left({C}\right)={\mathrm{Hom}}_{𝑓}\left({D}\right)$
75 74 ad4antr ${⊢}\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\to {\mathrm{Hom}}_{𝑓}\left({C}\right)={\mathrm{Hom}}_{𝑓}\left({D}\right)$
76 75 ad4antr ${⊢}\left(\left(\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\wedge {h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {\mathrm{Hom}}_{𝑓}\left({C}\right)={\mathrm{Hom}}_{𝑓}\left({D}\right)$
77 2 ad5antr ${⊢}\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\to {\mathrm{comp}}_{𝑓}\left({C}\right)={\mathrm{comp}}_{𝑓}\left({D}\right)$
78 77 ad4antr ${⊢}\left(\left(\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\wedge {h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {\mathrm{comp}}_{𝑓}\left({C}\right)={\mathrm{comp}}_{𝑓}\left({D}\right)$
79 simpllr ${⊢}\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\to {x}\in {\mathrm{Base}}_{{C}}$
80 79 ad2antrr ${⊢}\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\to {x}\in {\mathrm{Base}}_{{C}}$
81 80 ad4antr ${⊢}\left(\left(\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\wedge {h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {x}\in {\mathrm{Base}}_{{C}}$
82 simp-4r ${⊢}\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\to {y}\in {\mathrm{Base}}_{{C}}$
83 82 ad4antr ${⊢}\left(\left(\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\wedge {h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {y}\in {\mathrm{Base}}_{{C}}$
84 simpllr ${⊢}\left(\left(\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\wedge {h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {w}\in {\mathrm{Base}}_{{C}}$
85 simplr ${⊢}\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\to {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)$
86 85 ad4antr ${⊢}\left(\left(\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\wedge {h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)$
87 simpr ${⊢}\left(\left(\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\wedge {h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)$
88 70 71 72 73 76 78 81 83 84 86 87 comfeqval ${⊢}\left(\left(\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\wedge {h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to \left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}=\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}$
89 simpllr ${⊢}\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\to {z}\in {\mathrm{Base}}_{{C}}$
90 89 ad4antr ${⊢}\left(\left(\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\wedge {h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {z}\in {\mathrm{Base}}_{{C}}$
91 simp-4r ${⊢}\left(\left(\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\wedge {h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)$
92 simplr ${⊢}\left(\left(\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\wedge {h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)$
93 70 71 72 73 76 78 81 90 84 91 92 comfeqval ${⊢}\left(\left(\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\wedge {h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)$
94 88 93 eqeq12d ${⊢}\left(\left(\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\wedge {h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to \left(\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)↔\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)$
95 94 ex ${⊢}\left(\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\to \left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\to \left(\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)↔\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)$
96 95 ralimdva ${⊢}\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\to \left(\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\to \forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left(\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)↔\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)$
97 ralbi ${⊢}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left(\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)↔\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\to \left(\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)↔\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)$
98 96 97 syl6 ${⊢}\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\to \left(\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\to \left(\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)↔\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)$
99 98 ralimdva ${⊢}\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\to \left(\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\to \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)↔\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)$
100 99 impancom ${⊢}\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to \left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\to \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)↔\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)$
101 100 impr ${⊢}\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge \left(\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\right)\to \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)↔\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)$
102 ralbi ${⊢}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)↔\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\to \left(\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)↔\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)$
103 101 102 syl ${⊢}\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge \left(\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\right)\to \left(\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)↔\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)$
104 103 anbi2d ${⊢}\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge \left(\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\right)\to \left(\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)$
105 104 ex ${⊢}\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\to \left(\left(\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\to \left(\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
106 105 ralimdva ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\to \left(\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left(\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\to \forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left(\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
107 69 106 syl5bir ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\to \left(\left(\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge \forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\to \forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left(\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
108 107 expdimp ${⊢}\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge \forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to \left(\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\to \forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left(\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
109 ralbi ${⊢}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left(\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\to \left(\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)$
110 108 109 syl6 ${⊢}\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge \forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to \left(\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\to \left(\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
111 110 an32s ${⊢}\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge \forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\to \left(\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\to \left(\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
112 111 ralimdva ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge \forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to \left(\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\to \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\left(\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
113 ralbi ${⊢}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\left(\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\to \left(\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)$
114 112 113 syl6 ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge \forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to \left(\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\to \left(\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
115 114 expimpd ${⊢}\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\to \left(\left(\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\to \left(\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
116 115 ralimdva ${⊢}\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\to \left(\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\to \forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
117 ralbi ${⊢}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\to \left(\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)$
118 116 117 syl6 ${⊢}\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\to \left(\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\to \left(\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
119 68 118 syl5bir ${⊢}\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\to \left(\left(\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge \forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\to \left(\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
120 119 ralimdva ${⊢}\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\to \left(\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge \forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\to \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
121 ralbi ${⊢}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\to \left(\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)$
122 120 121 syl6 ${⊢}\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\to \left(\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge \forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\to \left(\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
123 67 122 syl5bir ${⊢}\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\to \left(\left(\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\to \left(\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
124 123 imp ${⊢}\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge \left(\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\right)\to \left(\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)$
125 124 an4s ${⊢}\left(\left({\phi }\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\wedge \left({x}\in {\mathrm{Base}}_{{C}}\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\right)\to \left(\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)$
126 125 anbi2d ${⊢}\left(\left({\phi }\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\wedge \left({x}\in {\mathrm{Base}}_{{C}}\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\right)\to \left(\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)↔\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
127 126 expr ${⊢}\left(\left({\phi }\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\to \left(\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\to \left(\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)↔\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)\right)$
128 127 ralimdva ${⊢}\left({\phi }\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\right)\to \left(\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\to \forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)↔\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)\right)$
129 128 expimpd ${⊢}{\phi }\to \left(\left(\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}{h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\in \left({y}\mathrm{Hom}\left({C}\right){w}\right)\wedge \forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\right)\to \forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)↔\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)\right)$
130 ralbi ${⊢}\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)↔\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)\to \left(\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)↔\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
131 66 129 130 syl56 ${⊢}{\phi }\to \left(\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\to \left(\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)↔\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)\right)$
132 10 16 131 pm5.21ndd ${⊢}{\phi }\to \left(\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)↔\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
133 1 homfeqbas ${⊢}{\phi }\to {\mathrm{Base}}_{{C}}={\mathrm{Base}}_{{D}}$
134 eqid ${⊢}\mathrm{Hom}\left({D}\right)=\mathrm{Hom}\left({D}\right)$
135 simpr ${⊢}\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\to {x}\in {\mathrm{Base}}_{{C}}$
136 70 71 134 74 135 135 homfeqval ${⊢}\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\to {x}\mathrm{Hom}\left({C}\right){x}={x}\mathrm{Hom}\left({D}\right){x}$
137 133 ad2antrr ${⊢}\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\to {\mathrm{Base}}_{{C}}={\mathrm{Base}}_{{D}}$
138 74 ad2antrr ${⊢}\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\to {\mathrm{Hom}}_{𝑓}\left({C}\right)={\mathrm{Hom}}_{𝑓}\left({D}\right)$
139 simpr ${⊢}\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\to {y}\in {\mathrm{Base}}_{{C}}$
140 simpllr ${⊢}\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\to {x}\in {\mathrm{Base}}_{{C}}$
141 70 71 134 138 139 140 homfeqval ${⊢}\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\to {y}\mathrm{Hom}\left({C}\right){x}={y}\mathrm{Hom}\left({D}\right){x}$
142 1 ad4antr ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\right)\to {\mathrm{Hom}}_{𝑓}\left({C}\right)={\mathrm{Hom}}_{𝑓}\left({D}\right)$
143 2 ad4antr ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\right)\to {\mathrm{comp}}_{𝑓}\left({C}\right)={\mathrm{comp}}_{𝑓}\left({D}\right)$
144 simplr ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\right)\to {y}\in {\mathrm{Base}}_{{C}}$
145 simp-4r ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\right)\to {x}\in {\mathrm{Base}}_{{C}}$
146 simpr ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\right)\to {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)$
147 simpllr ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\right)\to {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)$
148 70 71 72 73 142 143 144 145 145 146 147 comfeqval ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\right)\to {g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={g}\left(⟨{y},{x}⟩\mathrm{comp}\left({D}\right){x}\right){f}$
149 148 eqeq1d ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\right)\to \left({g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}↔{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({D}\right){x}\right){f}={f}\right)$
150 141 149 raleqbidva ${⊢}\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\to \left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}↔\forall {f}\in \left({y}\mathrm{Hom}\left({D}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({D}\right){x}\right){f}={f}\right)$
151 70 71 134 138 140 139 homfeqval ${⊢}\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\to {x}\mathrm{Hom}\left({C}\right){y}={x}\mathrm{Hom}\left({D}\right){y}$
152 1 ad4antr ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\to {\mathrm{Hom}}_{𝑓}\left({C}\right)={\mathrm{Hom}}_{𝑓}\left({D}\right)$
153 2 ad4antr ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\to {\mathrm{comp}}_{𝑓}\left({C}\right)={\mathrm{comp}}_{𝑓}\left({D}\right)$
154 simp-4r ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\to {x}\in {\mathrm{Base}}_{{C}}$
155 simplr ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\to {y}\in {\mathrm{Base}}_{{C}}$
156 simpllr ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\to {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)$
157 simpr ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\to {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)$
158 70 71 72 73 152 153 154 154 155 156 157 comfeqval ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\to {f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\left(⟨{x},{x}⟩\mathrm{comp}\left({D}\right){y}\right){g}$
159 158 eqeq1d ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\to \left({f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}↔{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({D}\right){y}\right){g}={f}\right)$
160 151 159 raleqbidva ${⊢}\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\to \left(\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}↔\forall {f}\in \left({x}\mathrm{Hom}\left({D}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({D}\right){y}\right){g}={f}\right)$
161 150 160 anbi12d ${⊢}\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\to \left(\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)↔\left(\forall {f}\in \left({y}\mathrm{Hom}\left({D}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({D}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({D}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({D}\right){y}\right){g}={f}\right)\right)$
162 137 161 raleqbidva ${⊢}\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\right)\to \left(\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)↔\forall {y}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({D}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({D}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({D}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({D}\right){y}\right){g}={f}\right)\right)$
163 136 162 rexeqbidva ${⊢}\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\to \left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)↔\exists {g}\in \left({x}\mathrm{Hom}\left({D}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({D}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({D}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({D}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({D}\right){y}\right){g}={f}\right)\right)$
164 133 adantr ${⊢}\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\to {\mathrm{Base}}_{{C}}={\mathrm{Base}}_{{D}}$
165 164 adantr ${⊢}\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\to {\mathrm{Base}}_{{C}}={\mathrm{Base}}_{{D}}$
166 74 ad2antrr ${⊢}\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\to {\mathrm{Hom}}_{𝑓}\left({C}\right)={\mathrm{Hom}}_{𝑓}\left({D}\right)$
167 simplr ${⊢}\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\to {y}\in {\mathrm{Base}}_{{C}}$
168 70 71 134 166 79 167 homfeqval ${⊢}\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\to {x}\mathrm{Hom}\left({C}\right){y}={x}\mathrm{Hom}\left({D}\right){y}$
169 simpr ${⊢}\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\to {z}\in {\mathrm{Base}}_{{C}}$
170 70 71 134 166 167 169 homfeqval ${⊢}\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\to {y}\mathrm{Hom}\left({C}\right){z}={y}\mathrm{Hom}\left({D}\right){z}$
171 170 adantr ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\to {y}\mathrm{Hom}\left({C}\right){z}={y}\mathrm{Hom}\left({D}\right){z}$
172 simpr ${⊢}\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\to {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)$
173 70 71 72 73 75 77 80 82 89 85 172 comfeqval ${⊢}\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\to {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}={g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}$
174 70 71 134 166 79 169 homfeqval ${⊢}\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\to {x}\mathrm{Hom}\left({C}\right){z}={x}\mathrm{Hom}\left({D}\right){z}$
175 174 ad2antrr ${⊢}\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\to {x}\mathrm{Hom}\left({C}\right){z}={x}\mathrm{Hom}\left({D}\right){z}$
176 173 175 eleq12d ${⊢}\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\to \left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)↔{g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({D}\right){z}\right)\right)$
177 164 ad4antr ${⊢}\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\to {\mathrm{Base}}_{{C}}={\mathrm{Base}}_{{D}}$
178 75 adantr ${⊢}\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\to {\mathrm{Hom}}_{𝑓}\left({C}\right)={\mathrm{Hom}}_{𝑓}\left({D}\right)$
179 simp-4r ${⊢}\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\to {z}\in {\mathrm{Base}}_{{C}}$
180 simpr ${⊢}\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\to {w}\in {\mathrm{Base}}_{{C}}$
181 70 71 134 178 179 180 homfeqval ${⊢}\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\to {z}\mathrm{Hom}\left({C}\right){w}={z}\mathrm{Hom}\left({D}\right){w}$
182 166 ad4antr ${⊢}\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {\mathrm{Hom}}_{𝑓}\left({C}\right)={\mathrm{Hom}}_{𝑓}\left({D}\right)$
183 2 ad7antr ${⊢}\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {\mathrm{comp}}_{𝑓}\left({C}\right)={\mathrm{comp}}_{𝑓}\left({D}\right)$
184 167 ad4antr ${⊢}\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {y}\in {\mathrm{Base}}_{{C}}$
185 169 ad4antr ${⊢}\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {z}\in {\mathrm{Base}}_{{C}}$
186 simplr ${⊢}\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {w}\in {\mathrm{Base}}_{{C}}$
187 simpllr ${⊢}\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)$
188 simpr ${⊢}\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)$
189 70 71 72 73 182 183 184 185 186 187 188 comfeqval ${⊢}\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}={h}\left(⟨{y},{z}⟩\mathrm{comp}\left({D}\right){w}\right){g}$
190 189 oveq1d ${⊢}\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\to \left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}=\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({D}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}$
191 79 ad4antr ${⊢}\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {x}\in {\mathrm{Base}}_{{C}}$
192 simp-4r ${⊢}\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)$
193 70 71 72 73 182 183 191 184 185 192 187 comfeqval ${⊢}\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}={g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}$
194 193 oveq2d ${⊢}\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\to {h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\right)$
195 190 194 eqeq12d ${⊢}\left(\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\wedge {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\right)\to \left(\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)↔\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({D}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\right)\right)$
196 181 195 raleqbidva ${⊢}\left(\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\wedge {w}\in {\mathrm{Base}}_{{C}}\right)\to \left(\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)↔\forall {h}\in \left({z}\mathrm{Hom}\left({D}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({D}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\right)\right)$
197 177 196 raleqbidva ${⊢}\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\to \left(\forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)↔\forall {w}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({D}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({D}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\right)\right)$
198 176 197 anbi12d ${⊢}\left(\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\wedge {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\right)\to \left(\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({D}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({D}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({D}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\right)\right)\right)$
199 171 198 raleqbidva ${⊢}\left(\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\wedge {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\right)\to \left(\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {g}\in \left({y}\mathrm{Hom}\left({D}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({D}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({D}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({D}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\right)\right)\right)$
200 168 199 raleqbidva ${⊢}\left(\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\wedge {z}\in {\mathrm{Base}}_{{C}}\right)\to \left(\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {f}\in \left({x}\mathrm{Hom}\left({D}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({D}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({D}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({D}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({D}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\right)\right)\right)$
201 165 200 raleqbidva ${⊢}\left(\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\wedge {y}\in {\mathrm{Base}}_{{C}}\right)\to \left(\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {z}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({D}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({D}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({D}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({D}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({D}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\right)\right)\right)$
202 164 201 raleqbidva ${⊢}\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\to \left(\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)↔\forall {y}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({D}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({D}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({D}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({D}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({D}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\right)\right)\right)$
203 163 202 anbi12d ${⊢}\left({\phi }\wedge {x}\in {\mathrm{Base}}_{{C}}\right)\to \left(\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)↔\left(\exists {g}\in \left({x}\mathrm{Hom}\left({D}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({D}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({D}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({D}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({D}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({D}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({D}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({D}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({D}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({D}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\right)\right)\right)\right)$
204 133 203 raleqbidva ${⊢}{\phi }\to \left(\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)↔\forall {x}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({D}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({D}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({D}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({D}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({D}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({D}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({D}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({D}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({D}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({D}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\right)\right)\right)\right)$
205 132 204 bitrd ${⊢}{\phi }\to \left(\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)↔\forall {x}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({D}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({D}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({D}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({D}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({D}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({D}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({D}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({D}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({D}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({D}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\right)\right)\right)\right)$
206 70 71 72 iscat ${⊢}{C}\in {V}\to \left({C}\in \mathrm{Cat}↔\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
207 3 206 syl ${⊢}{\phi }\to \left({C}\in \mathrm{Cat}↔\forall {x}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({C}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({C}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({C}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({C}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({C}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({C}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{C}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({C}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({C}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({C}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({C}\right){z}\right){f}\right)\right)\right)\right)$
208 eqid ${⊢}{\mathrm{Base}}_{{D}}={\mathrm{Base}}_{{D}}$
209 208 134 73 iscat ${⊢}{D}\in {W}\to \left({D}\in \mathrm{Cat}↔\forall {x}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({D}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({D}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({D}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({D}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({D}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({D}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({D}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({D}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({D}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({D}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\right)\right)\right)\right)$
210 4 209 syl ${⊢}{\phi }\to \left({D}\in \mathrm{Cat}↔\forall {x}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\left(\exists {g}\in \left({x}\mathrm{Hom}\left({D}\right){x}\right)\phantom{\rule{.4em}{0ex}}\forall {y}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\left(\forall {f}\in \left({y}\mathrm{Hom}\left({D}\right){x}\right)\phantom{\rule{.4em}{0ex}}{g}\left(⟨{y},{x}⟩\mathrm{comp}\left({D}\right){x}\right){f}={f}\wedge \forall {f}\in \left({x}\mathrm{Hom}\left({D}\right){y}\right)\phantom{\rule{.4em}{0ex}}{f}\left(⟨{x},{x}⟩\mathrm{comp}\left({D}\right){y}\right){g}={f}\right)\wedge \forall {y}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {z}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {f}\in \left({x}\mathrm{Hom}\left({D}\right){y}\right)\phantom{\rule{.4em}{0ex}}\forall {g}\in \left({y}\mathrm{Hom}\left({D}\right){z}\right)\phantom{\rule{.4em}{0ex}}\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\in \left({x}\mathrm{Hom}\left({D}\right){z}\right)\wedge \forall {w}\in {\mathrm{Base}}_{{D}}\phantom{\rule{.4em}{0ex}}\forall {h}\in \left({z}\mathrm{Hom}\left({D}\right){w}\right)\phantom{\rule{.4em}{0ex}}\left({h}\left(⟨{y},{z}⟩\mathrm{comp}\left({D}\right){w}\right){g}\right)\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){w}\right){f}={h}\left(⟨{x},{z}⟩\mathrm{comp}\left({D}\right){w}\right)\left({g}\left(⟨{x},{y}⟩\mathrm{comp}\left({D}\right){z}\right){f}\right)\right)\right)\right)$
211 205 207 210 3bitr4d ${⊢}{\phi }\to \left({C}\in \mathrm{Cat}↔{D}\in \mathrm{Cat}\right)$