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Metamath Proof Explorer

Theorem fthres2c

Description: Condition for a faithful functor to also be a faithful functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017)

Ref Expression
Hypotheses fthres2c.a 
|- A = ( Base ` C )
fthres2c.e 
|- E = ( D |`s S )
fthres2c.d 
|- ( ph -> D e. Cat )
fthres2c.r 
|- ( ph -> S e. V )
fthres2c.1 
|- ( ph -> F : A --> S )
Assertion fthres2c 
|- ( ph -> ( F ( C Faith D ) G <-> F ( C Faith E ) G ) )

Proof

Step Hyp Ref Expression
1 fthres2c.a 
 |-  A = ( Base ` C )
2 fthres2c.e 
 |-  E = ( D |`s S )
3 fthres2c.d 
 |-  ( ph -> D e. Cat )
4 fthres2c.r 
 |-  ( ph -> S e. V )
5 fthres2c.1 
 |-  ( ph -> F : A --> S )
6 1 2 3 4 5 funcres2c 
 |-  ( ph -> ( F ( C Func D ) G <-> F ( C Func E ) G ) )
7 6 anbi1d 
 |-  ( ph -> ( ( F ( C Func D ) G /\ A. x e. A A. y e. A Fun `' ( x G y ) ) <-> ( F ( C Func E ) G /\ A. x e. A A. y e. A Fun `' ( x G y ) ) ) )
8 1 isfth 
 |-  ( F ( C Faith D ) G <-> ( F ( C Func D ) G /\ A. x e. A A. y e. A Fun `' ( x G y ) ) )
9 1 isfth 
 |-  ( F ( C Faith E ) G <-> ( F ( C Func E ) G /\ A. x e. A A. y e. A Fun `' ( x G y ) ) )
10 7 8 9 3bitr4g 
 |-  ( ph -> ( F ( C Faith D ) G <-> F ( C Faith E ) G ) )