Theorem ffthres2c

Description: Condition for a fully faithful functor to also be a fully faithful functor into the restriction. (Contributed by Mario Carneiro, 27-Jan-2017)

	Ref	Expression
Hypotheses	ffthres2c.a	|- A = ( Base ` C )
	ffthres2c.e	|- E = ( D |`s S )
	ffthres2c.d	|- ( ph -> D e. Cat )
	ffthres2c.r	|- ( ph -> S e. V )
	ffthres2c.1	|- ( ph -> F : A --> S )
Assertion	ffthres2c	|- ( ph -> ( F ( ( C Full D ) i^i ( C Faith D ) ) G <-> F ( ( C Full E ) i^i ( C Faith E ) ) G ) )

Proof

Step	Hyp	Ref	Expression
1		ffthres2c.a	|- A = ( Base ` C )
2		ffthres2c.e	|- E = ( D |`s S )
3		ffthres2c.d	|- ( ph -> D e. Cat )
4		ffthres2c.r	|- ( ph -> S e. V )
5		ffthres2c.1	|- ( ph -> F : A --> S )
6	1 2 3 4 5	fullres2c	|- ( ph -> ( F ( C Full D ) G <-> F ( C Full E ) G ) )
7	1 2 3 4 5	fthres2c	|- ( ph -> ( F ( C Faith D ) G <-> F ( C Faith E ) G ) )
8	6 7	anbi12d	|- ( ph -> ( ( F ( C Full D ) G /\ F ( C Faith D ) G ) <-> ( F ( C Full E ) G /\ F ( C Faith E ) G ) ) )
9		brin	|- ( F ( ( C Full D ) i^i ( C Faith D ) ) G <-> ( F ( C Full D ) G /\ F ( C Faith D ) G ) )
10		brin	|- ( F ( ( C Full E ) i^i ( C Faith E ) ) G <-> ( F ( C Full E ) G /\ F ( C Faith E ) G ) )
11	8 9 10	3bitr4g	|- ( ph -> ( F ( ( C Full D ) i^i ( C Faith D ) ) G <-> F ( ( C Full E ) i^i ( C Faith E ) ) G ) )