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Theorem elixpconst 6703
Description: Membership in an infinite Cartesian product of a constant  B. (Contributed by NM, 12-Apr-2008.)
Hypothesis
Ref Expression
elixp.1  |-  F  e. 
_V
Assertion
Ref Expression
elixpconst  |-  ( F  e.  X_ x  e.  A  B 
<->  F : A --> B )
Distinct variable groups:    x, F    x, A    x, B

Proof of Theorem elixpconst
StepHypRef Expression
1 elixp.1 . . 3  |-  F  e. 
_V
21elixp 6702 . 2  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
3 ffnfv 5673 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B
) )
42, 3bitr4i 187 1  |-  ( F  e.  X_ x  e.  A  B 
<->  F : A --> B )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    e. wcel 2148   A.wral 2455   _Vcvv 2737    Fn wfn 5210   -->wf 5211   ` cfv 5215   X_cixp 6695
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-fv 5223  df-ixp 6696
This theorem is referenced by:  ixpconstg  6704
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