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Theorem mapval2 6449
Description: Alternate expression for the value of set exponentiation. (Contributed by NM, 3-Nov-2007.)
Hypotheses
Ref Expression
elmap.1  |-  A  e. 
_V
elmap.2  |-  B  e. 
_V
Assertion
Ref Expression
mapval2  |-  ( A  ^m  B )  =  ( ~P ( B  X.  A )  i^i 
{ f  |  f  Fn  B } )
Distinct variable group:    B, f
Allowed substitution hint:    A( f)

Proof of Theorem mapval2
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 dff2 5457 . . . 4  |-  ( g : B --> A  <->  ( g  Fn  B  /\  g  C_  ( B  X.  A
) ) )
2 ancom 263 . . . 4  |-  ( ( g  Fn  B  /\  g  C_  ( B  X.  A ) )  <->  ( g  C_  ( B  X.  A
)  /\  g  Fn  B ) )
31, 2bitri 183 . . 3  |-  ( g : B --> A  <->  ( g  C_  ( B  X.  A
)  /\  g  Fn  B ) )
4 elmap.1 . . . 4  |-  A  e. 
_V
5 elmap.2 . . . 4  |-  B  e. 
_V
64, 5elmap 6448 . . 3  |-  ( g  e.  ( A  ^m  B )  <->  g : B
--> A )
7 elin 3184 . . . 4  |-  ( g  e.  ( ~P ( B  X.  A )  i^i 
{ f  |  f  Fn  B } )  <-> 
( g  e.  ~P ( B  X.  A
)  /\  g  e.  { f  |  f  Fn  B } ) )
8 selpw 3440 . . . . 5  |-  ( g  e.  ~P ( B  X.  A )  <->  g  C_  ( B  X.  A
) )
9 vex 2623 . . . . . 6  |-  g  e. 
_V
10 fneq1 5115 . . . . . 6  |-  ( f  =  g  ->  (
f  Fn  B  <->  g  Fn  B ) )
119, 10elab 2761 . . . . 5  |-  ( g  e.  { f  |  f  Fn  B }  <->  g  Fn  B )
128, 11anbi12i 449 . . . 4  |-  ( ( g  e.  ~P ( B  X.  A )  /\  g  e.  { f  |  f  Fn  B } )  <->  ( g  C_  ( B  X.  A
)  /\  g  Fn  B ) )
137, 12bitri 183 . . 3  |-  ( g  e.  ( ~P ( B  X.  A )  i^i 
{ f  |  f  Fn  B } )  <-> 
( g  C_  ( B  X.  A )  /\  g  Fn  B )
)
143, 6, 133bitr4i 211 . 2  |-  ( g  e.  ( A  ^m  B )  <->  g  e.  ( ~P ( B  X.  A )  i^i  {
f  |  f  Fn  B } ) )
1514eqriv 2086 1  |-  ( A  ^m  B )  =  ( ~P ( B  X.  A )  i^i 
{ f  |  f  Fn  B } )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1290    e. wcel 1439   {cab 2075   _Vcvv 2620    i^i cin 2999    C_ wss 3000   ~Pcpw 3433    X. cxp 4450    Fn wfn 5023   -->wf 5024  (class class class)co 5666    ^m cmap 6419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-map 6421
This theorem is referenced by: (None)
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