Theorem fvixp 6762

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

Hypothesis

Ref	Expression
fvixp.1	|- ( x = C -> B = D )

Assertion

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

Distinct variable groups:   x, A   x, C   x, D   x, F

Allowed substitution hint:   B( x)

Proof of Theorem fvixp

Step	Hyp	Ref	Expression
1		elixp2 6761	. . 3 |- ( F e. X_ x e. A B <-> ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) )
2	1	simp3bi 1016	. 2 |- ( F e. X_ x e. A B -> A. x e. A ( F ` x ) e. B )
3		fveq2 5558	. . . 4 |- ( x = C -> ( F ` x ) = ( F ` C ) )
4		fvixp.1	. . . 4 |- ( x = C -> B = D )
5	3, 4	eleq12d 2267	. . 3 |- ( x = C -> ( ( F ` x ) e. B <-> ( F ` C ) e. D ) )
6	5	rspccva 2867	. 2 |- ( ( A. x e. A ( F ` x ) e. B /\ C e. A ) -> ( F ` C ) e. D )
7	2, 6	sylan 283	1 |- ( ( F e. X_ x e. A B /\ C e. A ) -> ( F ` C ) e. D )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   A.wral 2475   _Vcvv 2763   Fn wfn 5253   ` cfv 5258   X_cixp 6757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ixp 6758

This theorem is referenced by: (None)