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 = ( CatCat ` U )
	catcbas.b	|- B = ( Base ` C )
	catcbas.u	|- ( ph -> U e. V )
	catchomfval.h	|- H = ( Hom ` C )
	catchom.x	|- ( ph -> X e. B )
	catchom.y	|- ( ph -> Y e. B )
Assertion	catchom	|- ( ph -> ( X H Y ) = ( X Func Y ) )

Proof

Step	Hyp	Ref	Expression
1		catcbas.c	|- C = ( CatCat ` U )
2		catcbas.b	|- B = ( Base ` C )
3		catcbas.u	|- ( ph -> U e. V )
4		catchomfval.h	|- H = ( Hom ` C )
5		catchom.x	|- ( ph -> X e. B )
6		catchom.y	|- ( ph -> Y e. B )
7	1 2 3 4	catchomfval	|- ( ph -> H = ( x e. B , y e. B |-> ( x Func y ) ) )
8		oveq12	|- ( ( x = X /\ y = Y ) -> ( x Func y ) = ( X Func Y ) )
9	8	adantl	|- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( x Func y ) = ( X Func Y ) )
10		ovexd	|- ( ph -> ( X Func Y ) e. _V )
11	7 9 5 6 10	ovmpod	|- ( ph -> ( X H Y ) = ( X Func Y ) )