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

Theorem fnpm 7018
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 7013 . 2  |-  ^pm  =  ( x  e.  _V ,  y  e.  _V  |->  { f  e.  ~P ( y  X.  x
)  |  Fun  f } )
2 vex 2951 . . . . 5  |-  y  e. 
_V
3 vex 2951 . . . . 5  |-  x  e. 
_V
42, 3xpex 4982 . . . 4  |-  ( y  X.  x )  e. 
_V
54pwex 4374 . . 3  |-  ~P (
y  X.  x )  e.  _V
65rabex 4346 . 2  |-  { f  e.  ~P ( y  X.  x )  |  Fun  f }  e.  _V
71, 6fnmpt2i 6412 1  |-  ^pm  Fn  ( _V  X.  _V )
Colors of variables: wff set class
Syntax hints:   {crab 2701   _Vcvv 2948   ~Pcpw 3791    X. cxp 4868   Fun wfun 5440    Fn wfn 5441    ^pm cpm 7011
This theorem is referenced by:  elpmi  7027  pmresg  7033  pmsspw  7040
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-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  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-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-pm 7013
  Copyright terms: Public domain W3C validator