Theorem 2oppffunc

Description: The opposite functor of an opposite functor is 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 )
	2oppffunc.f	|- G = ( oppFunc ` F )
	2oppffunc.g	|- ( ph -> G e. ( O Func P ) )
Assertion	2oppffunc	|- ( ph -> ( oppFunc ` G ) 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	|- G = ( oppFunc ` F )
6		2oppffunc.g	|- ( ph -> G e. ( O Func P ) )
7		relfunc	|- Rel ( O Func P )
8	6 7 5	2oppf	|- ( ph -> ( oppFunc ` G ) = F )
9	5 6	eqeltrrid	|- ( ph -> ( oppFunc ` F ) e. ( O Func P ) )
10	1 2 3 4 9	funcoppc5	|- ( ph -> F e. ( C Func D ) )
11	8 10	eqeltrd	|- ( ph -> ( oppFunc ` G ) e. ( C Func D ) )