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 ) )