Theorem elixp 6917

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 6914	. 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 1009	. . 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 949	. 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 1005   e. wcel 2202   A.wral 2511   _Vcvv 2803   Fn wfn 5328   ` cfv 5333   X_cixp 6910

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 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-ixp 6911

This theorem is referenced by:  elixpconst  6918  ixpin  6935  ixpiinm  6936  elixpsn  6947