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 ) )