Theorem funcoppc5

Description: A functor on opposite categories yields a functor on the original categories. (Contributed by Zhi Wang, 14-Nov-2025)

	Ref	Expression
Hypotheses	funcoppc2.o	|- O = ( oppCat ` C )
	funcoppc2.p	|- P = ( oppCat ` D )
	funcoppc2.c	|- ( ph -> C e. V )
	funcoppc2.d	|- ( ph -> D e. W )
	funcoppc5.f	|- ( ph -> ( oppFunc ` F ) e. ( O Func P ) )
Assertion	funcoppc5	|- ( ph -> F e. ( C Func D ) )

Proof

Step	Hyp	Ref	Expression
1		funcoppc2.o	|- O = ( oppCat ` C )
2		funcoppc2.p	|- P = ( oppCat ` D )
3		funcoppc2.c	|- ( ph -> C e. V )
4		funcoppc2.d	|- ( ph -> D e. W )
5		funcoppc5.f	|- ( ph -> ( oppFunc ` F ) e. ( O Func P ) )
6		relfunc	|- Rel ( O Func P )
7		eqid	|- ( oppFunc ` F ) = ( oppFunc ` F )
8	5 6 7	oppfrcl	|- ( ph -> F e. ( _V X. _V ) )
9		1st2nd2	|- ( F e. ( _V X. _V ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
10	8 9	syl	|- ( ph -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
11	10	fveq2d	|- ( ph -> ( oppFunc ` F ) = ( oppFunc ` <. ( 1st ` F ) , ( 2nd ` F ) >. ) )
12		df-ov	|- ( ( 1st ` F ) oppFunc ( 2nd ` F ) ) = ( oppFunc ` <. ( 1st ` F ) , ( 2nd ` F ) >. )
13	11 12	eqtr4di	|- ( ph -> ( oppFunc ` F ) = ( ( 1st ` F ) oppFunc ( 2nd ` F ) ) )
14	13 5	eqeltrrd	|- ( ph -> ( ( 1st ` F ) oppFunc ( 2nd ` F ) ) e. ( O Func P ) )
15	1 2 3 4 14	funcoppc4	|- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
16		df-br	|- ( ( 1st ` F ) ( C Func D ) ( 2nd ` F ) <-> <. ( 1st ` F ) , ( 2nd ` F ) >. e. ( C Func D ) )
17	15 16	sylib	|- ( ph -> <. ( 1st ` F ) , ( 2nd ` F ) >. e. ( C Func D ) )
18	10 17	eqeltrd	|- ( ph -> F e. ( C Func D ) )