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Theorem mapval2 7467
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 6044 . . . 4  |-  ( g : B --> A  <->  ( g  Fn  B  /\  g  C_  ( B  X.  A
) ) )
2 ancom 450 . . . 4  |-  ( ( g  Fn  B  /\  g  C_  ( B  X.  A ) )  <->  ( g  C_  ( B  X.  A
)  /\  g  Fn  B ) )
31, 2bitri 249 . . 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 7466 . . 3  |-  ( g  e.  ( A  ^m  B )  <->  g : B
--> A )
7 elin 3683 . . . 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 4022 . . . . 5  |-  ( g  e.  ~P ( B  X.  A )  <->  g  C_  ( B  X.  A
) )
9 vex 3112 . . . . . 6  |-  g  e. 
_V
10 fneq1 5675 . . . . . 6  |-  ( f  =  g  ->  (
f  Fn  B  <->  g  Fn  B ) )
119, 10elab 3246 . . . . 5  |-  ( g  e.  { f  |  f  Fn  B }  <->  g  Fn  B )
128, 11anbi12i 697 . . . 4  |-  ( ( g  e.  ~P ( B  X.  A )  /\  g  e.  { f  |  f  Fn  B } )  <->  ( g  C_  ( B  X.  A
)  /\  g  Fn  B ) )
137, 12bitri 249 . . 3  |-  ( g  e.  ( ~P ( B  X.  A )  i^i 
{ f  |  f  Fn  B } )  <-> 
( g  C_  ( B  X.  A )  /\  g  Fn  B )
)
143, 6, 133bitr4i 277 . 2  |-  ( g  e.  ( A  ^m  B )  <->  g  e.  ( ~P ( B  X.  A )  i^i  {
f  |  f  Fn  B } ) )
1514eqriv 2453 1  |-  ( A  ^m  B )  =  ( ~P ( B  X.  A )  i^i 
{ f  |  f  Fn  B } )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442   _Vcvv 3109    i^i cin 3470    C_ wss 3471   ~Pcpw 4015    X. cxp 5006    Fn wfn 5589   -->wf 5590  (class class class)co 6296    ^m cmap 7438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440
This theorem is referenced by: (None)
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