Theorem sectss

Description: The section relation is a relation between morphisms from X to Y and morphisms from Y to X . (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Hypotheses	issect.b	$$⊢B={\mathrm{Base}}_C$$
	issect.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
	issect.o	$$⊢\underset{\dot{}}{·}=\mathrm{comp}\left(C\right)$$
	issect.i	$$⊢\underset{\dot{}}{1}=\mathrm{Id}\left(C\right)$$
	issect.s	$$⊢S=\mathrm{Sect}\left(C\right)$$
	issect.c	$$⊢\phi \to C\in \mathrm{Cat}$$
	issect.x	$$⊢\phi \to X\in B$$
	issect.y	$$⊢\phi \to Y\in B$$
Assertion	sectss	$$⊢\phi \to XSY\subseteq \left(XHY\right)\times \left(YHX\right)$$

Proof

Step	Hyp	Ref	Expression
1		issect.b	$$⊢B={\mathrm{Base}}_C$$
2		issect.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
3		issect.o	$$⊢\underset{\dot{}}{·}=\mathrm{comp}\left(C\right)$$
4		issect.i	$$⊢\underset{\dot{}}{1}=\mathrm{Id}\left(C\right)$$
5		issect.s	$$⊢S=\mathrm{Sect}\left(C\right)$$
6		issect.c	$$⊢\phi \to C\in \mathrm{Cat}$$
7		issect.x	$$⊢\phi \to X\in B$$
8		issect.y	$$⊢\phi \to Y\in B$$
9	1 2 3 4 5 6 7 8	sectfval	$$⊢\phi \to XSY=\left\{⟨f,g⟩|\left(\left(f\in \left(XHY\right)\wedge g\in \left(YHX\right)\right)\wedge g\left(⟨X,Y⟩\underset{\dot{}}{·}X\right)f=\underset{\dot{}}{1}\left(X\right)\right)\right\}$$
10		opabssxp	$$⊢\left\{⟨f,g⟩|\left(\left(f\in \left(XHY\right)\wedge g\in \left(YHX\right)\right)\wedge g\left(⟨X,Y⟩\underset{\dot{}}{·}X\right)f=\underset{\dot{}}{1}\left(X\right)\right)\right\}\subseteq \left(XHY\right)\times \left(YHX\right)$$
11	9 10	eqsstrdi	$$⊢\phi \to XSY\subseteq \left(XHY\right)\times \left(YHX\right)$$