Theorem catcrcl2

Description: Reverse closure for the category of categories (in a universe) (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 ) )
	catcrcl2.b	|- B = ( Base ` C )
Assertion	catcrcl2	|- ( ph -> ( X e. B /\ Y e. B ) )

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		catcrcl2.b	|- B = ( Base ` C )
5	1 2 3	catcrcl	|- ( ph -> U e. _V )
6	1 4 5 2	catchomfval	|- ( ph -> H = ( x e. B , y e. B |-> ( x Func y ) ) )
7	6	oveqd	|- ( ph -> ( X H Y ) = ( X ( x e. B , y e. B |-> ( x Func y ) ) Y ) )
8	3 7	eleqtrd	|- ( ph -> F e. ( X ( x e. B , y e. B |-> ( x Func y ) ) Y ) )
9		eqid	|- ( x e. B , y e. B |-> ( x Func y ) ) = ( x e. B , y e. B |-> ( x Func y ) )
10	9	elmpocl	|- ( F e. ( X ( x e. B , y e. B |-> ( x Func y ) ) Y ) -> ( X e. B /\ Y e. B ) )
11	8 10	syl	|- ( ph -> ( X e. B /\ Y e. B ) )