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Theorem mapvalg 6987
Description: The value of set exponentiation.  ( 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.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
mapvalg  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  ^m  B
)  =  { f  |  f : B --> A } )
Distinct variable groups:    A, f    B, f
Allowed substitution hints:    C( f)    D( f)

Proof of Theorem mapvalg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mapex 6983 . . 3  |-  ( ( B  e.  D  /\  A  e.  C )  ->  { f  |  f : B --> A }  e.  _V )
21ancoms 440 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  f : B --> A }  e.  _V )
3 elex 2924 . . 3  |-  ( A  e.  C  ->  A  e.  _V )
4 elex 2924 . . 3  |-  ( B  e.  D  ->  B  e.  _V )
5 feq3 5537 . . . . . 6  |-  ( x  =  A  ->  (
f : y --> x  <-> 
f : y --> A ) )
65abbidv 2518 . . . . 5  |-  ( x  =  A  ->  { f  |  f : y --> x }  =  {
f  |  f : y --> A } )
7 feq2 5536 . . . . . 6  |-  ( y  =  B  ->  (
f : y --> A  <-> 
f : B --> A ) )
87abbidv 2518 . . . . 5  |-  ( y  =  B  ->  { f  |  f : y --> A }  =  {
f  |  f : B --> A } )
9 df-map 6979 . . . . 5  |-  ^m  =  ( x  e.  _V ,  y  e.  _V  |->  { f  |  f : y --> x }
)
106, 8, 9ovmpt2g 6167 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  {
f  |  f : B --> A }  e.  _V )  ->  ( A  ^m  B )  =  { f  |  f : B --> A }
)
11103expia 1155 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { f  |  f : B --> A }  e.  _V  ->  ( A  ^m  B )  =  {
f  |  f : B --> A } ) )
123, 4, 11syl2an 464 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { f  |  f : B --> A }  e.  _V  ->  ( A  ^m  B )  =  {
f  |  f : B --> A } ) )
132, 12mpd 15 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  ^m  B
)  =  { f  |  f : B --> A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   _Vcvv 2916   -->wf 5409  (class class class)co 6040    ^m cmap 6977
This theorem is referenced by:  mapval  6989  elmapg  6990  ixpconstg  7030  hashf1lem2  11660  birthdaylem1  20743  birthdaylem2  20744  cnfex  27566
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-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-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-map 6979
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