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Theorem elixp2 7544
Description: Membership in an infinite Cartesian product. See df-ixp 7541 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 5674 . . . . 5  |-  ( f  =  F  ->  (
f  Fn  A  <->  F  Fn  A ) )
2 fveq1 5878 . . . . . . 7  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
32eleq1d 2533 . . . . . 6  |-  ( f  =  F  ->  (
( f `  x
)  e.  B  <->  ( F `  x )  e.  B
) )
43ralbidv 2829 . . . . 5  |-  ( f  =  F  ->  ( A. x  e.  A  ( f `  x
)  e.  B  <->  A. x  e.  A  ( F `  x )  e.  B
) )
51, 4anbi12d 725 . . . 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 7542 . . . 4  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B ) }
75, 6elab2g 3175 . . 3  |-  ( F  e.  _V  ->  ( F  e.  X_ x  e.  A  B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B
) ) )
87pm5.32i 649 . 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 3040 . . 3  |-  ( F  e.  X_ x  e.  A  B  ->  F  e.  _V )
109pm4.71ri 645 . 2  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  e.  _V  /\  F  e.  X_ x  e.  A  B )
)
11 3anass 1011 . 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 285 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031    Fn wfn 5584   ` cfv 5589   X_cixp 7540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597  df-ixp 7541
This theorem is referenced by:  fvixp  7545  ixpfn  7546  elixp  7547  ixpf  7562  resixp  7575  undifixp  7576  mptelixpg  7577  prdsbasprj  15448  xpsfrnel  15547  isssc  15803  isfuncd  15848  funcres2b  15880  dprdw  17720  ptrecube  32004  kelac1  35992  fvixp2  37549  hoiqssbllem1  38562
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