Theorem 2oppffunc

Description: The opposite functor of an opposite functor is a functor on the original categories. (Contributed by Zhi Wang, 14-Nov-2025) The functor in opposite categories does not have to be an opposite functor. (Revised by Zhi Wang, 17-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 )
	2oppffunc.f	|- ( ph -> F e. ( O Func P ) )
Assertion	2oppffunc	|- ( ph -> ( oppFunc ` 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		2oppffunc.f	|- ( ph -> F e. ( O Func P ) )
6		oppfval2	|- ( F e. ( O Func P ) -> ( oppFunc ` F ) = <. ( 1st ` F ) , tpos ( 2nd ` F ) >. )
7	5 6	syl	|- ( ph -> ( oppFunc ` F ) = <. ( 1st ` F ) , tpos ( 2nd ` F ) >. )
8	5	func1st2nd	|- ( ph -> ( 1st ` F ) ( O Func P ) ( 2nd ` F ) )
9	1 2 3 4 8	funcoppc2	|- ( ph -> ( 1st ` F ) ( C Func D ) tpos ( 2nd ` F ) )
10		df-br	|- ( ( 1st ` F ) ( C Func D ) tpos ( 2nd ` F ) <-> <. ( 1st ` F ) , tpos ( 2nd ` F ) >. e. ( C Func D ) )
11	9 10	sylib	|- ( ph -> <. ( 1st ` F ) , tpos ( 2nd ` F ) >. e. ( C Func D ) )
12	7 11	eqeltrd	|- ( ph -> ( oppFunc ` F ) e. ( C Func D ) )