Theorem catchom

Description: Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017)

	Ref	Expression
Hypotheses	catcbas.c	$$⊢C=\mathrm{CatCat}\left(U\right)$$
	catcbas.b	$$⊢B={\mathrm{Base}}_C$$
	catcbas.u	$$⊢\phi \to U\in V$$
	catchomfval.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
	catchom.x	$$⊢\phi \to X\in B$$
	catchom.y	$$⊢\phi \to Y\in B$$
Assertion	catchom	$$⊢\phi \to XHY=X\mathrm{Func}Y$$

Proof

Step	Hyp	Ref	Expression
1		catcbas.c	$$⊢C=\mathrm{CatCat}\left(U\right)$$
2		catcbas.b	$$⊢B={\mathrm{Base}}_C$$
3		catcbas.u	$$⊢\phi \to U\in V$$
4		catchomfval.h	$$⊢H=\mathrm{Hom}\left(C\right)$$
5		catchom.x	$$⊢\phi \to X\in B$$
6		catchom.y	$$⊢\phi \to Y\in B$$
7	1 2 3 4	catchomfval	$$⊢\phi \to H=\left(x\in B,y\in B⟼x\mathrm{Func}y\right)$$
8		oveq12	$$⊢\left(x=X\wedge y=Y\right)\to x\mathrm{Func}y=X\mathrm{Func}Y$$
9	8	adantl	$$⊢\left(\phi \wedge \left(x=X\wedge y=Y\right)\right)\to x\mathrm{Func}y=X\mathrm{Func}Y$$
10		ovexd	$$⊢\phi \to X\mathrm{Func}Y\in \mathrm{V}$$
11	7 9 5 6 10	ovmpod	$$⊢\phi \to XHY=X\mathrm{Func}Y$$