Theorem funcoppc3

Description: A functor on opposite categories yields a functor on the original categories. (Contributed by Zhi Wang, 4-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	$$⊢\phi \to D\in W$$
	funcoppc3.f	$$⊢\phi \to F\left(O\mathrm{Func}P\right)tposG$$
	funcoppc3.g	$$⊢\phi \to GFn\left(A\times B\right)$$
Assertion	funcoppc3	$$⊢\phi \to F\left(C\mathrm{Func}D\right)G$$

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		funcoppc3.f	$$⊢\phi \to F\left(O\mathrm{Func}P\right)tposG$$
6		funcoppc3.g	$$⊢\phi \to GFn\left(A\times B\right)$$
7	1 2 3 4 5	funcoppc2	$$⊢\phi \to F\left(C\mathrm{Func}D\right)tpostposG$$
8		fnrel	$$⊢GFn\left(A\times B\right)\to \mathrm{Rel}G$$
9	6 8	syl	$$⊢\phi \to \mathrm{Rel}G$$
10		relxp	$$⊢\mathrm{Rel}\left(A\times B\right)$$
11	6	fndmd	$$⊢\phi \to \mathrm{dom}G=A\times B$$
12	11	releqd	$$⊢\phi \to \left(\mathrm{Rel}\mathrm{dom}G\leftrightarrow \mathrm{Rel}\left(A\times B\right)\right)$$
13	10 12	mpbiri	$$⊢\phi \to \mathrm{Rel}\mathrm{dom}G$$
14		tpostpos2	$$⊢\left(\mathrm{Rel}G\wedge \mathrm{Rel}\mathrm{dom}G\right)\to tpostposG=G$$
15	9 13 14	syl2anc	$$⊢\phi \to tpostposG=G$$
16	7 15	breqtrd	$$⊢\phi \to F\left(C\mathrm{Func}D\right)G$$