MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnpm Unicode version

Theorem fnpm 6964
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 6959 . 2  |-  ^pm  =  ( x  e.  _V ,  y  e.  _V  |->  { f  e.  ~P ( y  X.  x
)  |  Fun  f } )
2 vex 2904 . . . . 5  |-  y  e. 
_V
3 vex 2904 . . . . 5  |-  x  e. 
_V
42, 3xpex 4932 . . . 4  |-  ( y  X.  x )  e. 
_V
54pwex 4325 . . 3  |-  ~P (
y  X.  x )  e.  _V
65rabex 4297 . 2  |-  { f  e.  ~P ( y  X.  x )  |  Fun  f }  e.  _V
71, 6fnmpt2i 6361 1  |-  ^pm  Fn  ( _V  X.  _V )
Colors of variables: wff set class
Syntax hints:   {crab 2655   _Vcvv 2901   ~Pcpw 3744    X. cxp 4818   Fun wfun 5390    Fn wfn 5391    ^pm cpm 6957
This theorem is referenced by:  elpmi  6973  pmresg  6979  pmsspw  6986
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 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-pm 6959
  Copyright terms: Public domain W3C validator