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Theorem mapval 6800
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 6798 . 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 1632    e. wcel 1696   {cab 2282   _Vcvv 2801   -->wf 5267  (class class class)co 5874    ^m cmap 6788
This theorem is referenced by:  lautset  30893  pautsetN  30909  tendoset  31570
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790
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