MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fabex Unicode version

Theorem fabex 5439
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 5438 . 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 1632    e. wcel 1696   {cab 2282   _Vcvv 2801   -->wf 5267
This theorem is referenced by:  isoriso  25315
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
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-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275
  Copyright terms: Public domain W3C validator