Theorem fvixp 6813

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 6812	. . 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 1017	. 2 |- ( F e. X_ x e. A B -> A. x e. A ( F ` x ) e. B )
3		fveq2 5599	. . . 4 |- ( x = C -> ( F ` x ) = ( F ` C ) )
4		fvixp.1	. . . 4 |- ( x = C -> B = D )
5	3, 4	eleq12d 2278	. . 3 |- ( x = C -> ( ( F ` x ) e. B <-> ( F ` C ) e. D ) )
6	5	rspccva 2883	. 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 1373   e. wcel 2178   A.wral 2486   _Vcvv 2776   Fn wfn 5285   ` cfv 5290   X_cixp 6808

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-ixp 6809

This theorem is referenced by: (None)