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Theorem elixpconst 6840
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 6839 . 2  |-  ( F  e.  X_ x  e.  A  B 
<->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B ) )
3 ffnfv 5701 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  A. x  e.  A  ( F `  x )  e.  B
) )
42, 3bitr4i 243 1  |-  ( F  e.  X_ x  e.  A  B 
<->  F : A --> B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1696   A.wral 2556   _Vcvv 2801    Fn wfn 5266   -->wf 5267   ` cfv 5271   X_cixp 6833
This theorem is referenced by:  ixpconstg  6841  sscpwex  13708  psrbaglefi  16134
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ixp 6834
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