Theorem fthoppf

Description: The opposite functor of a faithful functor is also faithful. Proposition 3.43(c) in Adamek p. 39. (Contributed by Zhi Wang, 26-Nov-2025)

	Ref	Expression
Hypotheses	fulloppf.o	$$⊢O=\mathrm{oppCat}\left(C\right)$$
	fulloppf.p	$$⊢P=\mathrm{oppCat}\left(D\right)$$
	fthoppf.f	$$⊢\phi \to F\in \left(C\mathrm{Faith}D\right)$$
Assertion	fthoppf	Could not format assertion : No typesetting found for |- ( ph -> ( oppFunc ` F ) e. ( O Faith P ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		fulloppf.o	$$⊢O=\mathrm{oppCat}\left(C\right)$$
2		fulloppf.p	$$⊢P=\mathrm{oppCat}\left(D\right)$$
3		fthoppf.f	$$⊢\phi \to F\in \left(C\mathrm{Faith}D\right)$$
4		fthfunc	$$⊢C\mathrm{Faith}D\subseteq C\mathrm{Func}D$$
5	4	sseli	$$⊢F\in \left(C\mathrm{Faith}D\right)\to F\in \left(C\mathrm{Func}D\right)$$
6		oppfval2	Could not format ( F e. ( C Func D ) -> ( oppFunc ` F ) = <. ( 1st ` F ) , tpos ( 2nd ` F ) >. ) : No typesetting found for |- ( F e. ( C Func D ) -> ( oppFunc ` F ) = <. ( 1st ` F ) , tpos ( 2nd ` F ) >. ) with typecode |-
7	3 5 6	3syl	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		relfth	$$⊢\mathrm{Rel}\left(C\mathrm{Faith}D\right)$$
9		1st2ndbr	$$⊢\left(\mathrm{Rel}\left(C\mathrm{Faith}D\right)\wedge F\in \left(C\mathrm{Faith}D\right)\right)\to 1^{st}\left(F\right)\left(C\mathrm{Faith}D\right)2^{nd}\left(F\right)$$
10	8 3 9	sylancr	$$⊢\phi \to 1^{st}\left(F\right)\left(C\mathrm{Faith}D\right)2^{nd}\left(F\right)$$
11	1 2 10	fthoppc	$$⊢\phi \to 1^{st}\left(F\right)\left(O\mathrm{Faith}P\right)tpos{2}^{nd}\left(F\right)$$
12		df-br	$$⊢1^{st}\left(F\right)\left(O\mathrm{Faith}P\right)tpos{2}^{nd}\left(F\right)\leftrightarrow ⟨1^{st}\left(F\right),tpos{2}^{nd}\left(F\right)⟩\in \left(O\mathrm{Faith}P\right)$$
13	11 12	sylib	$$⊢\phi \to ⟨1^{st}\left(F\right),tpos{2}^{nd}\left(F\right)⟩\in \left(O\mathrm{Faith}P\right)$$
14	7 13	eqeltrd	Could not format ( ph -> ( oppFunc ` F ) e. ( O Faith P ) ) : No typesetting found for |- ( ph -> ( oppFunc ` F ) e. ( O Faith P ) ) with typecode |-