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Theorem mapvalg 6777
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
Dummy variables  x  y are mutually distinct and distinct from all other variables.
Allowed substitution hints:    C( f)    D( f)

Proof of Theorem mapvalg
StepHypRef Expression
1 mapex 6773 . . 3  |-  ( ( B  e.  D  /\  A  e.  C )  ->  { f  |  f : B --> A }  e.  _V )
21ancoms 441 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  { f  |  f : B --> A }  e.  _V )
3 elex 2797 . . 3  |-  ( A  e.  C  ->  A  e.  _V )
4 elex 2797 . . 3  |-  ( B  e.  D  ->  B  e.  _V )
5 feq3 5342 . . . . . 6  |-  ( x  =  A  ->  (
f : y --> x  <-> 
f : y --> A ) )
65abbidv 2398 . . . . 5  |-  ( x  =  A  ->  { f  |  f : y --> x }  =  {
f  |  f : y --> A } )
7 feq2 5341 . . . . . 6  |-  ( y  =  B  ->  (
f : y --> A  <-> 
f : B --> A ) )
87abbidv 2398 . . . . 5  |-  ( y  =  B  ->  { f  |  f : y --> A }  =  {
f  |  f : B --> A } )
9 df-map 6769 . . . . 5  |-  ^m  =  ( x  e.  _V ,  y  e.  _V  |->  { f  |  f : y --> x }
)
106, 8, 9ovmpt2g 5943 . . . 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 465 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( { f  |  f : B --> A }  e.  _V  ->  ( A  ^m  B )  =  {
f  |  f : B --> A } ) )
132, 12mpd 16 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A  ^m  B
)  =  { f  |  f : B --> A } )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685   {cab 2270   _Vcvv 2789   -->wf 5217  (class class class)co 5819    ^m cmap 6767
This theorem is referenced by:  mapval  6779  elmapg  6780  ixpconstg  6820  hashf1lem2  11388  birthdaylem1  20240  birthdaylem2  20241  mapex2  24539  cnfex  27098
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-map 6769
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