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Theorem pmex 7015
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 2547 . 2  |-  { f  |  ( Fun  f  /\  f  C_  ( A  X.  B ) ) }  =  { f  |  ( f  C_  ( A  X.  B
)  /\  Fun  f ) }
3 xpexg 4981 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
4 abssexg 4376 . . 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 2519 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 1725   {cab 2421   _Vcvv 2948    C_ wss 3312    X. cxp 4868   Fun wfun 5440
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-13 1727  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-pow 4369  ax-pr 4395  ax-un 4693
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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rex 2703  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-opab 4259  df-xp 4876
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