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Theorem pmex 6959
Description: The class of all partial functions from one set to another is a set. (Contributed by NM, 15-Nov-2007.)
Assertion
Ref Expression
pmex  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  ( Fun  f  /\  f  C_  ( A  X.  B
) ) }  e.  _V )
Distinct variable groups:    A, f    B, f
Allowed substitution hints:    C( f)    D( f)

Proof of Theorem pmex
StepHypRef Expression
1 ancom 438 . . 3  |-  ( ( Fun  f  /\  f  C_  ( A  X.  B
) )  <->  ( f  C_  ( A  X.  B
)  /\  Fun  f ) )
21abbii 2499 . 2  |-  { f  |  ( Fun  f  /\  f  C_  ( A  X.  B ) ) }  =  { f  |  ( f  C_  ( A  X.  B
)  /\  Fun  f ) }
3 xpexg 4929 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
4 abssexg 4325 . . 3  |-  ( ( A  X.  B )  e.  _V  ->  { f  |  ( f  C_  ( A  X.  B
)  /\  Fun  f ) }  e.  _V )
53, 4syl 16 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  ( f  C_  ( A  X.  B )  /\  Fun  f ) }  e.  _V )
62, 5syl5eqel 2471 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  ( Fun  f  /\  f  C_  ( A  X.  B
) ) }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   {cab 2373   _Vcvv 2899    C_ wss 3263    X. cxp 4816   Fun wfun 5388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-rex 2655  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-opab 4208  df-xp 4824
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