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Theorem elixpconst 7062
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 7061 . 2  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
3 ffnfv 5886 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B
) )
42, 3bitr4i 244 1  |-  ( F  e.  X_ x  e.  A  B 
<->  F : A --> B )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1725   A.wral 2697   _Vcvv 2948    Fn wfn 5441   -->wf 5442   ` cfv 5446   X_cixp 7055
This theorem is referenced by:  ixpconstg  7063  sscpwex  14007  psrbaglefi  16429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ixp 7056
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