Theorem elixp 6940

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 6937	. 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 2203   A.wral 2520   _Vcvv 2813   Fn wfn 5347   ` cfv 5352   X_cixp 6933

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 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-ixp 6934

This theorem is referenced by:  elixpconst  6941  ixpin  6958  ixpiinm  6959  elixpsn  6970