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Theorem elixp2 7259
Description: Membership in an infinite Cartesian product. See df-ixp 7256 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 5494 . . . . 5  |-  ( f  =  F  ->  (
f  Fn  A  <->  F  Fn  A ) )
2 fveq1 5685 . . . . . . 7  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
32eleq1d 2504 . . . . . 6  |-  ( f  =  F  ->  (
( f `  x
)  e.  B  <->  ( F `  x )  e.  B
) )
43ralbidv 2730 . . . . 5  |-  ( f  =  F  ->  ( A. x  e.  A  ( f `  x
)  e.  B  <->  A. x  e.  A  ( F `  x )  e.  B
) )
51, 4anbi12d 710 . . . 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 7257 . . . 4  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B ) }
75, 6elab2g 3103 . . 3  |-  ( F  e.  _V  ->  ( F  e.  X_ x  e.  A  B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B
) ) )
87pm5.32i 637 . 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 2976 . . 3  |-  ( F  e.  X_ x  e.  A  B  ->  F  e.  _V )
109pm4.71ri 633 . 2  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  e.  _V  /\  F  e.  X_ x  e.  A  B )
)
11 3anass 969 . 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710   _Vcvv 2967    Fn wfn 5408   ` cfv 5413   X_cixp 7255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-iota 5376  df-fun 5415  df-fn 5416  df-fv 5421  df-ixp 7256
This theorem is referenced by:  fvixp  7260  ixpfn  7261  elixp  7262  ixpf  7277  resixp  7290  undifixp  7291  mptelixpg  7292  prdsbasprj  14402  xpsfrnel  14493  isssc  14725  isfuncd  14767  funcres2b  14799  dprdw  16482  dprdwOLD  16488  kelac1  29369
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