Theorem catcrcl

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 ) )
Assertion	catcrcl	|- ( ph -> U e. _V )

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		elfvne0	|- ( F e. ( H ` <. X , Y >. ) -> H =/= (/) )
5		df-ov	|- ( X H Y ) = ( H ` <. X , Y >. )
6	4 5	eleq2s	|- ( F e. ( X H Y ) -> H =/= (/) )
7		fvprc	|- ( -. U e. _V -> ( CatCat ` U ) = (/) )
8	1 7	eqtrid	|- ( -. U e. _V -> C = (/) )
9		fveq2	|- ( C = (/) -> ( Hom ` C ) = ( Hom ` (/) ) )
10		homid	|- Hom = Slot ( Hom ` ndx )
11	10	str0	|- (/) = ( Hom ` (/) )
12	9 2 11	3eqtr4g	|- ( C = (/) -> H = (/) )
13	8 12	syl	|- ( -. U e. _V -> H = (/) )
14	13	necon1ai	|- ( H =/= (/) -> U e. _V )
15	3 6 14	3syl	|- ( ph -> U e. _V )