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Theorem fnmap 6984
Description: Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
fnmap  |-  ^m  Fn  ( _V  X.  _V )

Proof of Theorem fnmap
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-map 6979 . 2  |-  ^m  =  ( x  e.  _V ,  y  e.  _V  |->  { f  |  f : y --> x }
)
2 vex 2919 . . 3  |-  y  e. 
_V
3 vex 2919 . . 3  |-  x  e. 
_V
4 mapex 6983 . . 3  |-  ( ( y  e.  _V  /\  x  e.  _V )  ->  { f  |  f : y --> x }  e.  _V )
52, 3, 4mp2an 654 . 2  |-  { f  |  f : y --> x }  e.  _V
61, 5fnmpt2i 6379 1  |-  ^m  Fn  ( _V  X.  _V )
Colors of variables: wff set class
Syntax hints:    e. wcel 1721   {cab 2390   _Vcvv 2916    X. cxp 4835    Fn wfn 5408   -->wf 5409    ^m cmap 6977
This theorem is referenced by:  elmapex  6996
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-map 6979
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