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Theorem elixp2 7369
Description: Membership in an infinite Cartesian product. See df-ixp 7366 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 5599 . . . . 5  |-  ( f  =  F  ->  (
f  Fn  A  <->  F  Fn  A ) )
2 fveq1 5790 . . . . . . 7  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
32eleq1d 2520 . . . . . 6  |-  ( f  =  F  ->  (
( f `  x
)  e.  B  <->  ( F `  x )  e.  B
) )
43ralbidv 2838 . . . . 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 7367 . . . 4  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B ) }
75, 6elab2g 3207 . . 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 3079 . . 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 1370    e. wcel 1758   A.wral 2795   _Vcvv 3070    Fn wfn 5513   ` cfv 5518   X_cixp 7365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-iota 5481  df-fun 5520  df-fn 5521  df-fv 5526  df-ixp 7366
This theorem is referenced by:  fvixp  7370  ixpfn  7371  elixp  7372  ixpf  7387  resixp  7400  undifixp  7401  mptelixpg  7402  prdsbasprj  14514  xpsfrnel  14605  isssc  14837  isfuncd  14879  funcres2b  14911  dprdw  16601  dprdwOLD  16607  kelac1  29556
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