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Theorem mapvalg 6672
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 6668 . . 3  |-  ( ( B  e.  D  /\  A  e.  C )  ->  { f  |  f : B --> A }  e.  _V )
21ancoms 268 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  f : B --> A }  e.  _V )
3 elex 2760 . . 3  |-  ( A  e.  C  ->  A  e.  _V )
4 elex 2760 . . 3  |-  ( B  e.  D  ->  B  e.  _V )
5 feq3 5362 . . . . . 6  |-  ( x  =  A  ->  (
f : y --> x  <-> 
f : y --> A ) )
65abbidv 2305 . . . . 5  |-  ( x  =  A  ->  { f  |  f : y --> x }  =  {
f  |  f : y --> A } )
7 feq2 5361 . . . . . 6  |-  ( y  =  B  ->  (
f : y --> A  <-> 
f : B --> A ) )
87abbidv 2305 . . . . 5  |-  ( y  =  B  ->  { f  |  f : y --> A }  =  {
f  |  f : B --> A } )
9 df-map 6664 . . . . 5  |-  ^m  =  ( x  e.  _V ,  y  e.  _V  |->  { f  |  f : y --> x }
)
106, 8, 9ovmpog 6023 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  {
f  |  f : B --> A }  e.  _V )  ->  ( A  ^m  B )  =  { f  |  f : B --> A }
)
11103expia 1206 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { f  |  f : B --> A }  e.  _V  ->  ( A  ^m  B )  =  {
f  |  f : B --> A } ) )
123, 4, 11syl2an 289 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { f  |  f : B --> A }  e.  _V  ->  ( A  ^m  B )  =  {
f  |  f : B --> A } ) )
132, 12mpd 13 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 104    = wceq 1363    e. wcel 2158   {cab 2173   _Vcvv 2749   -->wf 5224  (class class class)co 5888    ^m cmap 6662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-map 6664
This theorem is referenced by:  mapval  6674  elmapg  6675  ixpconstg  6721  ptex  12731  cnovex  14049  ispsmet  14176  cncfval  14412
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