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Theorem fnpm 7506
Description: Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)
Assertion
Ref Expression
fnpm  |-  ^pm  Fn  ( _V  X.  _V )

Proof of Theorem fnpm
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pm 7501 . 2  |-  ^pm  =  ( x  e.  _V ,  y  e.  _V  |->  { f  e.  ~P ( y  X.  x
)  |  Fun  f } )
2 vex 3060 . . . . 5  |-  y  e. 
_V
3 vex 3060 . . . . 5  |-  x  e. 
_V
42, 3xpex 6622 . . . 4  |-  ( y  X.  x )  e. 
_V
54pwex 4600 . . 3  |-  ~P (
y  X.  x )  e.  _V
65rabex 4568 . 2  |-  { f  e.  ~P ( y  X.  x )  |  Fun  f }  e.  _V
71, 6fnmpt2i 6889 1  |-  ^pm  Fn  ( _V  X.  _V )
Colors of variables: wff setvar class
Syntax hints:   {crab 2753   _Vcvv 3057   ~Pcpw 3963    X. cxp 4851   Fun wfun 5595    Fn wfn 5596    ^pm cpm 7499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-oprab 6319  df-mpt2 6320  df-1st 6820  df-2nd 6821  df-pm 7501
This theorem is referenced by:  elpmi  7516  pmresg  7525  pmsspw  7532
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