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Theorem elixpconst 6593
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 6592 . 2  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
3 ffnfv 5571 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B
) )
42, 3bitr4i 186 1  |-  ( F  e.  X_ x  e.  A  B 
<->  F : A --> B )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    e. wcel 1480   A.wral 2414   _Vcvv 2681    Fn wfn 5113   -->wf 5114   ` cfv 5118   X_cixp 6585
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-ixp 6586
This theorem is referenced by:  ixpconstg  6594
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