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	$$⊢O=\mathrm{oppCat}\left(C\right)$$
	funcoppc2.p	$$⊢P=\mathrm{oppCat}\left(D\right)$$
	funcoppc2.c	$$⊢\phi \to C\in V$$
	funcoppc2.d	$$⊢\phi \to D\in W$$
	2oppffunc.f	$$⊢\phi \to F\in \left(O\mathrm{Func}P\right)$$
Assertion	2oppffunc	Could not format assertion : No typesetting found for |- ( ph -> ( oppFunc ` F ) e. ( C Func D ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		funcoppc2.o	$$⊢O=\mathrm{oppCat}\left(C\right)$$
2		funcoppc2.p	$$⊢P=\mathrm{oppCat}\left(D\right)$$
3		funcoppc2.c	$$⊢\phi \to C\in V$$
4		funcoppc2.d	$$⊢\phi \to D\in W$$
5		2oppffunc.f	$$⊢\phi \to F\in \left(O\mathrm{Func}P\right)$$
6		oppfval2	Could not format ( F e. ( O Func P ) -> ( oppFunc ` F ) = <. ( 1st ` F ) , tpos ( 2nd ` F ) >. ) : No typesetting found for |- ( F e. ( O Func P ) -> ( oppFunc ` F ) = <. ( 1st ` F ) , tpos ( 2nd ` F ) >. ) with typecode |-
7	5 6	syl	Could not format ( ph -> ( oppFunc ` F ) = <. ( 1st ` F ) , tpos ( 2nd ` F ) >. ) : No typesetting found for |- ( ph -> ( oppFunc ` F ) = <. ( 1st ` F ) , tpos ( 2nd ` F ) >. ) with typecode |-
8	5	func1st2nd	$$⊢\phi \to 1^{st}\left(F\right)\left(O\mathrm{Func}P\right)2^{nd}\left(F\right)$$
9	1 2 3 4 8	funcoppc2	$$⊢\phi \to 1^{st}\left(F\right)\left(C\mathrm{Func}D\right)tpos{2}^{nd}\left(F\right)$$
10		df-br	$$⊢1^{st}\left(F\right)\left(C\mathrm{Func}D\right)tpos{2}^{nd}\left(F\right)\leftrightarrow ⟨1^{st}\left(F\right),tpos{2}^{nd}\left(F\right)⟩\in \left(C\mathrm{Func}D\right)$$
11	9 10	sylib	$$⊢\phi \to ⟨1^{st}\left(F\right),tpos{2}^{nd}\left(F\right)⟩\in \left(C\mathrm{Func}D\right)$$
12	7 11	eqeltrd	Could not format ( ph -> ( oppFunc ` F ) e. ( C Func D ) ) : No typesetting found for |- ( ph -> ( oppFunc ` F ) e. ( C Func D ) ) with typecode |-