Theorem swapf2fval

Description: The morphism part of the swap functor. See also swapf2fvala . (Contributed by Zhi Wang, 7-Oct-2025)

	Ref	Expression
Hypotheses	swapfval.c	$$⊢\phi \to C\in U$$
	swapfval.d	$$⊢\phi \to D\in V$$
	swapf2fvala.s	$$⊢S=C{\times }_{c}D$$
	swapf2fvala.b	$$⊢B={\mathrm{Base}}_S$$
	swapf2fvala.h	$$⊢\phi \to H=\mathrm{Hom}\left(S\right)$$
	swapf2fval.o	No typesetting found for |- ( ph -> ( C swapF D ) = <. O , P >. ) with typecode |-
Assertion	swapf2fval	$$⊢\phi \to P=\left(u\in B,v\in B⟼\left(f\in \left(uHv\right)⟼\bigcup {\left\{f\right\}}^{-1}\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		swapfval.c	$$⊢\phi \to C\in U$$
2		swapfval.d	$$⊢\phi \to D\in V$$
3		swapf2fvala.s	$$⊢S=C{\times }_{c}D$$
4		swapf2fvala.b	$$⊢B={\mathrm{Base}}_S$$
5		swapf2fvala.h	$$⊢\phi \to H=\mathrm{Hom}\left(S\right)$$
6		swapf2fval.o	Could not format ( ph -> ( C swapF D ) = <. O , P >. ) : No typesetting found for |- ( ph -> ( C swapF D ) = <. O , P >. ) with typecode |-
7	6	fveq2d	Could not format ( ph -> ( 2nd ` ( C swapF D ) ) = ( 2nd ` <. O , P >. ) ) : No typesetting found for |- ( ph -> ( 2nd ` ( C swapF D ) ) = ( 2nd ` <. O , P >. ) ) with typecode |-
8	1 2 3 4 5	swapf2fvala	Could not format ( ph -> ( 2nd ` ( C swapF D ) ) = ( u e. B , v e. B |-> ( f e. ( u H v ) |-> U. `' { f } ) ) ) : No typesetting found for |- ( ph -> ( 2nd ` ( C swapF D ) ) = ( u e. B , v e. B |-> ( f e. ( u H v ) |-> U. `' { f } ) ) ) with typecode |-
9	1 2	swapfelvv	Could not format ( ph -> ( C swapF D ) e. ( _V X. _V ) ) : No typesetting found for |- ( ph -> ( C swapF D ) e. ( _V X. _V ) ) with typecode |-
10	6 9	eqeltrrd	$$⊢\phi \to ⟨O,P⟩\in \left(\mathrm{V}\times \mathrm{V}\right)$$
11		opelxp	$$⊢⟨O,P⟩\in \left(\mathrm{V}\times \mathrm{V}\right)\leftrightarrow \left(O\in \mathrm{V}\wedge P\in \mathrm{V}\right)$$
12	11	biimpi	$$⊢⟨O,P⟩\in \left(\mathrm{V}\times \mathrm{V}\right)\to \left(O\in \mathrm{V}\wedge P\in \mathrm{V}\right)$$
13		op2ndg	$$⊢\left(O\in \mathrm{V}\wedge P\in \mathrm{V}\right)\to 2^{nd}\left(⟨O,P⟩\right)=P$$
14	10 12 13	3syl	$$⊢\phi \to 2^{nd}\left(⟨O,P⟩\right)=P$$
15	7 8 14	3eqtr3rd	$$⊢\phi \to P=\left(u\in B,v\in B⟼\left(f\in \left(uHv\right)⟼\bigcup {\left\{f\right\}}^{-1}\right)\right)$$