This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem fthres2b

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

Ref Expression
Hypotheses fthres2b.a 
|- A = ( Base ` C )
fthres2b.h 
|- H = ( Hom ` C )
fthres2b.r 
|- ( ph -> R e. ( Subcat ` D ) )
fthres2b.s 
|- ( ph -> R Fn ( S X. S ) )
fthres2b.1 
|- ( ph -> F : A --> S )
fthres2b.2 
|- ( ( ph /\ ( x e. A /\ y e. A ) ) -> ( x G y ) : Y --> ( ( F ` x ) R ( F ` y ) ) )
Assertion fthres2b 
|- ( ph -> ( F ( C Faith D ) G <-> F ( C Faith ( D |`cat R ) ) G ) )

Proof

Step Hyp Ref Expression
1 fthres2b.a 
 |-  A = ( Base ` C )
2 fthres2b.h 
 |-  H = ( Hom ` C )
3 fthres2b.r 
 |-  ( ph -> R e. ( Subcat ` D ) )
4 fthres2b.s 
 |-  ( ph -> R Fn ( S X. S ) )
5 fthres2b.1 
 |-  ( ph -> F : A --> S )
6 fthres2b.2 
 |-  ( ( ph /\ ( x e. A /\ y e. A ) ) -> ( x G y ) : Y --> ( ( F ` x ) R ( F ` y ) ) )
7 1 2 3 4 5 6 funcres2b 
 |-  ( ph -> ( F ( C Func D ) G <-> F ( C Func ( D |`cat R ) ) G ) )
8 7 anbi1d 
 |-  ( ph -> ( ( F ( C Func D ) G /\ A. x e. A A. y e. A Fun `' ( x G y ) ) <-> ( F ( C Func ( D |`cat R ) ) G /\ A. x e. A A. y e. A Fun `' ( x G y ) ) ) )
9 1 isfth 
 |-  ( F ( C Faith D ) G <-> ( F ( C Func D ) G /\ A. x e. A A. y e. A Fun `' ( x G y ) ) )
10 1 isfth 
 |-  ( F ( C Faith ( D |`cat R ) ) G <-> ( F ( C Func ( D |`cat R ) ) G /\ A. x e. A A. y e. A Fun `' ( x G y ) ) )
11 8 9 10 3bitr4g 
 |-  ( ph -> ( F ( C Faith D ) G <-> F ( C Faith ( D |`cat R ) ) G ) )