Theorem fvixp

Description: Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016)

	Ref	Expression
Hypothesis	fvixp.1	|- ( x = C -> B = D )
Assertion	fvixp	|- ( ( F e. X_ x e. A B /\ C e. A ) -> ( F ` C ) e. D )

Proof

Step	Hyp	Ref	Expression
1		fvixp.1	|- ( x = C -> B = D )
2		elixp2	|- ( F e. X_ x e. A B <-> ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) )
3	2	simp3bi	|- ( F e. X_ x e. A B -> A. x e. A ( F ` x ) e. B )
4		fveq2	|- ( x = C -> ( F ` x ) = ( F ` C ) )
5	4 1	eleq12d	|- ( x = C -> ( ( F ` x ) e. B <-> ( F ` C ) e. D ) )
6	5	rspccva	|- ( ( A. x e. A ( F ` x ) e. B /\ C e. A ) -> ( F ` C ) e. D )
7	3 6	sylan	|- ( ( F e. X_ x e. A B /\ C e. A ) -> ( F ` C ) e. D )