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Theorem pmex 6793
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 437 . . 3  |-  ( ( Fun  f  /\  f  C_  ( A  X.  B
) )  <->  ( f  C_  ( A  X.  B
)  /\  Fun  f ) )
21abbii 2408 . 2  |-  { f  |  ( Fun  f  /\  f  C_  ( A  X.  B ) ) }  =  { f  |  ( f  C_  ( A  X.  B
)  /\  Fun  f ) }
3 xpexg 4816 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
4 abssexg 4211 . . 3  |-  ( ( A  X.  B )  e.  _V  ->  { f  |  ( f  C_  ( A  X.  B
)  /\  Fun  f ) }  e.  _V )
53, 4syl 15 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  ( f  C_  ( A  X.  B )  /\  Fun  f ) }  e.  _V )
62, 5syl5eqel 2380 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 358    e. wcel 1696   {cab 2282   _Vcvv 2801    C_ wss 3165    X. cxp 4703   Fun wfun 5265
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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rex 2562  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-opab 4094  df-xp 4711
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