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

Theorem mapval 6780
Description: The value of set exponentiation (inference version). 
( A  ^m  B
) is the set of all functions that map from  B to  A. Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.)
Hypotheses
Ref Expression
mapval.1  |-  A  e. 
_V
mapval.2  |-  B  e. 
_V
Assertion
Ref Expression
mapval  |-  ( A  ^m  B )  =  { f  |  f : B --> A }
Distinct variable groups:    A, f    B, f

Proof of Theorem mapval
StepHypRef Expression
1 mapval.1 . 2  |-  A  e. 
_V
2 mapval.2 . 2  |-  B  e. 
_V
3 mapvalg 6778 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  ^m  B
)  =  { f  |  f : B --> A } )
41, 2, 3mp2an 653 1  |-  ( A  ^m  B )  =  { f  |  f : B --> A }
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1685   {cab 2270   _Vcvv 2789   -->wf 5217  (class class class)co 5820    ^m cmap 6768
This theorem is referenced by:  lautset  29550  pautsetN  29566  tendoset  30227
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-map 6770
  Copyright terms: Public domain W3C validator