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Theorem fabex 5423
Description: Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
Hypotheses
Ref Expression
fabex.1  |-  A  e. 
_V
fabex.2  |-  B  e. 
_V
fabex.3  |-  F  =  { x  |  ( x : A --> B  /\  ph ) }
Assertion
Ref Expression
fabex  |-  F  e. 
_V
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    ph( x)    F( x)

Proof of Theorem fabex
StepHypRef Expression
1 fabex.1 . 2  |-  A  e. 
_V
2 fabex.2 . 2  |-  B  e. 
_V
3 fabex.3 . . 3  |-  F  =  { x  |  ( x : A --> B  /\  ph ) }
43fabexg 5422 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  F  e.  _V )
51, 2, 4mp2an 653 1  |-  F  e. 
_V
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788   -->wf 5251
This theorem is referenced by:  isoriso  25212
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259
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