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Theorem mapval 6966
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 6964 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  ^m  B
)  =  { f  |  f : B --> A } )
41, 2, 3mp2an 654 1  |-  ( A  ^m  B )  =  { f  |  f : B --> A }
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717   {cab 2373   _Vcvv 2899   -->wf 5390  (class class class)co 6020    ^m cmap 6954
This theorem is referenced by:  lautset  30196  pautsetN  30212  tendoset  30873
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 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-map 6956
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