Theorem fvixp 6759

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 6758	. . 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 5555	. . . 4 |- ( x = C -> ( F ` x ) = ( F ` C ) )
4		fvixp.1	. . . 4 |- ( x = C -> B = D )
5	3, 4	eleq12d 2264	. . 3 |- ( x = C -> ( ( F ` x ) e. B <-> ( F ` C ) e. D ) )
6	5	rspccva 2864	. 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 2164   A.wral 2472   _Vcvv 2760   Fn wfn 5250   ` cfv 5255   X_cixp 6754

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-ixp 6755

This theorem is referenced by: (None)