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Theorem pmex 7476
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 451 . . 3  |-  ( ( Fun  f  /\  f  C_  ( A  X.  B
) )  <->  ( f  C_  ( A  X.  B
)  /\  Fun  f ) )
21abbii 2554 . 2  |-  { f  |  ( Fun  f  /\  f  C_  ( A  X.  B ) ) }  =  { f  |  ( f  C_  ( A  X.  B
)  /\  Fun  f ) }
3 xpexg 6598 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  X.  B
)  e.  _V )
4 abssexg 4601 . . 3  |-  ( ( A  X.  B )  e.  _V  ->  { f  |  ( f  C_  ( A  X.  B
)  /\  Fun  f ) }  e.  _V )
53, 4syl 17 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  ( f  C_  ( A  X.  B )  /\  Fun  f ) }  e.  _V )
62, 5syl5eqel 2512 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  ( Fun  f  /\  f  C_  ( A  X.  B
) ) }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1867   {cab 2405   _Vcvv 3078    C_ wss 3433    X. cxp 4843   Fun wfun 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-opab 4476  df-xp 4851  df-rel 4852
This theorem is referenced by: (None)
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