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Theorem elixp2 6603
Description: Membership in an infinite Cartesian product. See df-ixp 6600 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 5218 . . . . 5  |-  ( f  =  F  ->  (
f  Fn  A  <->  F  Fn  A ) )
2 fveq1 5427 . . . . . . 7  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
32eleq1d 2209 . . . . . 6  |-  ( f  =  F  ->  (
( f `  x
)  e.  B  <->  ( F `  x )  e.  B
) )
43ralbidv 2438 . . . . 5  |-  ( f  =  F  ->  ( A. x  e.  A  ( f `  x
)  e.  B  <->  A. x  e.  A  ( F `  x )  e.  B
) )
51, 4anbi12d 465 . . . 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 6601 . . . 4  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B ) }
75, 6elab2g 2834 . . 3  |-  ( F  e.  _V  ->  ( F  e.  X_ x  e.  A  B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B
) ) )
87pm5.32i 450 . 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 2700 . . 3  |-  ( F  e.  X_ x  e.  A  B  ->  F  e.  _V )
109pm4.71ri 390 . 2  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  e.  _V  /\  F  e.  X_ x  e.  A  B )
)
11 3anass 967 . 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 211 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 set class
Syntax hints:    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1332    e. wcel 1481   A.wral 2417   _Vcvv 2689    Fn wfn 5125   ` cfv 5130   X_cixp 6599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fn 5133  df-fv 5138  df-ixp 6600
This theorem is referenced by:  fvixp  6604  ixpfn  6605  elixp  6606  ixpf  6621  resixp  6634  mptelixpg  6635
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