Theorem fvixp 6940

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 6939	. . 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 1041	. 2 |- ( F e. X_ x e. A B -> A. x e. A ( F ` x ) e. B )
3		fveq2 5672	. . . 4 |- ( x = C -> ( F ` x ) = ( F ` C ) )
4		fvixp.1	. . . 4 |- ( x = C -> B = D )
5	3, 4	eleq12d 2305	. . 3 |- ( x = C -> ( ( F ` x ) e. B <-> ( F ` C ) e. D ) )
6	5	rspccva 2922	. 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 1398   e. wcel 2205   A.wral 2522   _Vcvv 2815   Fn wfn 5349   ` cfv 5354   X_cixp 6935

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362  df-ixp 6936

This theorem is referenced by: (None)