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Metamath Proof Explorer

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 ) )