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Theorem mapvalg 7019
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 7015 . . 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 2956 . . 3  |-  ( A  e.  C  ->  A  e.  _V )
4 elex 2956 . . 3  |-  ( B  e.  D  ->  B  e.  _V )
5 feq3 5569 . . . . . 6  |-  ( x  =  A  ->  (
f : y --> x  <-> 
f : y --> A ) )
65abbidv 2549 . . . . 5  |-  ( x  =  A  ->  { f  |  f : y --> x }  =  {
f  |  f : y --> A } )
7 feq2 5568 . . . . . 6  |-  ( y  =  B  ->  (
f : y --> A  <-> 
f : B --> A ) )
87abbidv 2549 . . . . 5  |-  ( y  =  B  ->  { f  |  f : y --> A }  =  {
f  |  f : B --> A } )
9 df-map 7011 . . . . 5  |-  ^m  =  ( x  e.  _V ,  y  e.  _V  |->  { f  |  f : y --> x }
)
106, 8, 9ovmpt2g 6199 . . . 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 1652    e. wcel 1725   {cab 2421   _Vcvv 2948   -->wf 5441  (class class class)co 6072    ^m cmap 7009
This theorem is referenced by:  mapval  7021  elmapg  7022  ixpconstg  7062  hashf1lem2  11693  birthdaylem1  20778  birthdaylem2  20779  cnfex  27613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-map 7011
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