Theorem ffthoppc

Description: The opposite functor of a fully faithful functor is also full and faithful. (Contributed by Mario Carneiro, 27-Jan-2017)

	Ref	Expression
Hypotheses	fulloppc.o	$$⊢O=\mathrm{oppCat}\left(C\right)$$
	fulloppc.p	$$⊢P=\mathrm{oppCat}\left(D\right)$$
	ffthoppc.f	$$⊢\phi \to F\left(\left(C\mathrm{Full}D\right)\cap \left(C\mathrm{Faith}D\right)\right)G$$
Assertion	ffthoppc	$$⊢\phi \to F\left(\left(O\mathrm{Full}P\right)\cap \left(O\mathrm{Faith}P\right)\right)tposG$$

Proof

Step	Hyp	Ref	Expression
1		fulloppc.o	$$⊢O=\mathrm{oppCat}\left(C\right)$$
2		fulloppc.p	$$⊢P=\mathrm{oppCat}\left(D\right)$$
3		ffthoppc.f	$$⊢\phi \to F\left(\left(C\mathrm{Full}D\right)\cap \left(C\mathrm{Faith}D\right)\right)G$$
4		brin	$$⊢F\left(\left(C\mathrm{Full}D\right)\cap \left(C\mathrm{Faith}D\right)\right)G\leftrightarrow \left(F\left(C\mathrm{Full}D\right)G\wedge F\left(C\mathrm{Faith}D\right)G\right)$$
5	3 4	sylib	$$⊢\phi \to \left(F\left(C\mathrm{Full}D\right)G\wedge F\left(C\mathrm{Faith}D\right)G\right)$$
6	5	simpld	$$⊢\phi \to F\left(C\mathrm{Full}D\right)G$$
7	1 2 6	fulloppc	$$⊢\phi \to F\left(O\mathrm{Full}P\right)tposG$$
8	5	simprd	$$⊢\phi \to F\left(C\mathrm{Faith}D\right)G$$
9	1 2 8	fthoppc	$$⊢\phi \to F\left(O\mathrm{Faith}P\right)tposG$$
10		brin	$$⊢F\left(\left(O\mathrm{Full}P\right)\cap \left(O\mathrm{Faith}P\right)\right)tposG\leftrightarrow \left(F\left(O\mathrm{Full}P\right)tposG\wedge F\left(O\mathrm{Faith}P\right)tposG\right)$$
11	7 9 10	sylanbrc	$$⊢\phi \to F\left(\left(O\mathrm{Full}P\right)\cap \left(O\mathrm{Faith}P\right)\right)tposG$$