Theorem elcatchom

Description: A morphism of the category of categories (in a universe) is a functor. See df-catc for the definition of the category Cat, which consists of all categories in the universe u (i.e., " u -small categories", see Definition 3.44. of Adamek p. 39), with functors as the morphisms ( catchom ). (Contributed by Zhi Wang, 14-Nov-2025)

	Ref	Expression
Hypotheses	catcrcl.c	|- C = ( CatCat ` U )
	catcrcl.h	|- H = ( Hom ` C )
	catcrcl.f	|- ( ph -> F e. ( X H Y ) )
Assertion	elcatchom	|- ( ph -> F e. ( X Func Y ) )

Proof

Step	Hyp	Ref	Expression
1		catcrcl.c	|- C = ( CatCat ` U )
2		catcrcl.h	|- H = ( Hom ` C )
3		catcrcl.f	|- ( ph -> F e. ( X H Y ) )
4		eqid	|- ( Base ` C ) = ( Base ` C )
5	1 2 3	catcrcl	|- ( ph -> U e. _V )
6	1 2 3 4	catcrcl2	|- ( ph -> ( X e. ( Base ` C ) /\ Y e. ( Base ` C ) ) )
7	6	simpld	|- ( ph -> X e. ( Base ` C ) )
8	6	simprd	|- ( ph -> Y e. ( Base ` C ) )
9	1 4 5 2 7 8	catchom	|- ( ph -> ( X H Y ) = ( X Func Y ) )
10	3 9	eleqtrd	|- ( ph -> F e. ( X Func Y ) )