Theorem elixp 6815

Description: Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)

Hypothesis

Ref	Expression
elixp.1	|- F e. _V

Assertion

Ref	Expression
elixp	|- ( F e. X_ x e. A B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )

Distinct variable groups:   x, F   x, A

Allowed substitution hint:   B( x)

Proof of Theorem elixp

Step	Hyp	Ref	Expression
1		elixp2 6812	. 2 |- ( F e. X_ x e. A B <-> ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) )
2		elixp.1	. . 3 |- F e. _V
3		3anass 985	. . 3 |- ( ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) <-> ( F e. _V /\ ( F Fn A /\ A. x e. A ( F ` x ) e. B ) ) )
4	2, 3	mpbiran 943	. 2 |- ( ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. B ) <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )
5	1, 4	bitri 184	1 |- ( F e. X_ x e. A B <-> ( F Fn A /\ A. x e. A ( F ` x ) e. B ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 981   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:  elixpconst  6816  ixpin  6833  ixpiinm  6834  elixpsn  6845