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Theorem elixp 7072
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
StepHypRef Expression
1 elixp2 7069 . 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 941 . . 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 ) ) )
42, 3mpbiran 886 . 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
) )
51, 4bitri 242 1  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    /\ w3a 937    e. wcel 1726   A.wral 2707   _Vcvv 2958    Fn wfn 5452   ` cfv 5457   X_cixp 7066
This theorem is referenced by:  elixpconst  7073  ixpin  7090  ixpiin  7091  resixpfo  7103  elixpsn  7104  boxriin  7107  boxcutc  7108  ixpfi2  7408  ixpiunwdom  7562  dfac9  8021  ac9  8368  ac9s  8378  konigthlem  8448  xpscf  13796  cofucl  14090  yonedalem3  14382  psrbaglefi  16442  ptpjpre1  17608  ptpjcn  17648  ptpjopn  17649  ptclsg  17652  dfac14  17655  pthaus  17675  xkopt  17692  ptcmplem2  18089  ptcmplem3  18090  ptcmplem4  18091  prdsbl  18526  prdsxmslem2  18564  ptpcon  24925  inixp  26444  prdstotbnd  26517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465  df-ixp 7067
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