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Theorem elixp 6754
Description: Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)
Hypothesis
Ref Expression
elixp.1  |-  F  e. 
_V
Assertion
Ref Expression
elixp  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
Distinct variable groups:    x, F    x, A
Allowed substitution hint:    B( x)

Proof of Theorem elixp
StepHypRef Expression
1 elixp2 6751 . 2  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
2 elixp.1 . . 3  |-  F  e. 
_V
3 3anass 984 . . 3  |-  ( ( 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 ) ) )
42, 3mpbiran 942 . 2  |-  ( ( F  e.  _V  /\  F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B )  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B
) )
51, 4bitri 184 1  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    /\ w3a 980    e. wcel 2164   A.wral 2472   _Vcvv 2760    Fn wfn 5245   ` cfv 5250   X_cixp 6747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-rel 4664  df-cnv 4665  df-co 4666  df-dm 4667  df-iota 5211  df-fun 5252  df-fn 5253  df-fv 5258  df-ixp 6748
This theorem is referenced by:  elixpconst  6755  ixpin  6772  ixpiinm  6773  elixpsn  6784
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