Theorem fvixp 6978

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 6977	. . 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 1045	. 2 |- ( F e. X_ x e. A B -> A. x e. A ( F ` x ) e. B )
3		fveq2 5693	. . . 4 |- ( x = C -> ( F ` x ) = ( F ` C ) )
4		fvixp.1	. . . 4 |- ( x = C -> B = D )
5	3, 4	eleq12d 2309	. . 3 |- ( x = C -> ( ( F ` x ) e. B <-> ( F ` C ) e. D ) )
6	5	rspccva 2928	. 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 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   Fn wfn 5370   ` cfv 5375   X_cixp 6973

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-ixp 6974

This theorem is referenced by: (None)