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Theorem mapval 6786
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 6784 . 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 1625    e. wcel 1686   {cab 2271   _Vcvv 2790   -->wf 5253  (class class class)co 5860    ^m cmap 6774
This theorem is referenced by:  lautset  30344  pautsetN  30360  tendoset  31021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-map 6776
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