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Theorem mapvalg 6750
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
StepHypRef Expression
1 mapex 6746 . . 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 2771 . . 3  |-  ( A  e.  C  ->  A  e.  _V )
4 elex 2771 . . 3  |-  ( B  e.  D  ->  B  e.  _V )
5 feq3 5315 . . . . . 6  |-  ( x  =  A  ->  (
f : y --> x  <-> 
f : y --> A ) )
65abbidv 2372 . . . . 5  |-  ( x  =  A  ->  { f  |  f : y --> x }  =  {
f  |  f : y --> A } )
7 feq2 5314 . . . . . 6  |-  ( y  =  B  ->  (
f : y --> A  <-> 
f : B --> A ) )
87abbidv 2372 . . . . 5  |-  ( y  =  B  ->  { f  |  f : y --> A }  =  {
f  |  f : B --> A } )
9 df-map 6742 . . . . 5  |-  ^m  =  ( x  e.  _V ,  y  e.  _V  |->  { f  |  f : y --> x }
)
106, 8, 9ovmpt2g 5916 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  {
f  |  f : B --> A }  e.  _V )  ->  ( A  ^m  B )  =  { f  |  f : B --> A }
)
11103expia 1158 . . 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 1619    e. wcel 1621   {cab 2244   _Vcvv 2763   -->wf 4669  (class class class)co 5792    ^m cmap 6740
This theorem is referenced by:  mapval  6752  elmapg  6753  ixpconstg  6793  hashf1lem2  11359  birthdaylem1  20208  birthdaylem2  20209  mapex2  24507  cnfex  27067
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-map 6742
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