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Theorem elixp2 7466
Description: Membership in an infinite Cartesian product. See df-ixp 7463 for discussion of the notation. (Contributed by NM, 28-Sep-2006.)
Assertion
Ref Expression
elixp2  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
Distinct variable groups:    x, A    x, F
Allowed substitution hint:    B( x)

Proof of Theorem elixp2
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 fneq1 5651 . . . . 5  |-  ( f  =  F  ->  (
f  Fn  A  <->  F  Fn  A ) )
2 fveq1 5847 . . . . . . 7  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
32eleq1d 2523 . . . . . 6  |-  ( f  =  F  ->  (
( f `  x
)  e.  B  <->  ( F `  x )  e.  B
) )
43ralbidv 2893 . . . . 5  |-  ( f  =  F  ->  ( A. x  e.  A  ( f `  x
)  e.  B  <->  A. x  e.  A  ( F `  x )  e.  B
) )
51, 4anbi12d 708 . . . 4  |-  ( f  =  F  ->  (
( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B )  <-> 
( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) ) )
6 dfixp 7464 . . . 4  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B ) }
75, 6elab2g 3245 . . 3  |-  ( F  e.  _V  ->  ( F  e.  X_ x  e.  A  B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B
) ) )
87pm5.32i 635 . 2  |-  ( ( F  e.  _V  /\  F  e.  X_ x  e.  A  B )  <->  ( F  e.  _V  /\  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) ) )
9 elex 3115 . . 3  |-  ( F  e.  X_ x  e.  A  B  ->  F  e.  _V )
109pm4.71ri 631 . 2  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  e.  _V  /\  F  e.  X_ x  e.  A  B )
)
11 3anass 975 . 2  |-  ( ( 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 ) ) )
128, 10, 113bitr4i 277 1  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106    Fn wfn 5565   ` cfv 5570   X_cixp 7462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578  df-ixp 7463
This theorem is referenced by:  fvixp  7467  ixpfn  7468  elixp  7469  ixpf  7484  resixp  7497  undifixp  7498  mptelixpg  7499  prdsbasprj  14961  xpsfrnel  15052  isssc  15308  isfuncd  15353  funcres2b  15385  dprdw  17238  dprdwOLD  17245  kelac1  31248
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