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Theorem mapval2 7043
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 5881 . . . 4  |-  ( g : B --> A  <->  ( g  Fn  B  /\  g  C_  ( B  X.  A
) ) )
2 ancom 438 . . . 4  |-  ( ( g  Fn  B  /\  g  C_  ( B  X.  A ) )  <->  ( g  C_  ( B  X.  A
)  /\  g  Fn  B ) )
31, 2bitri 241 . . 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 7042 . . 3  |-  ( g  e.  ( A  ^m  B )  <->  g : B
--> A )
7 elin 3530 . . . 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 vex 2959 . . . . . 6  |-  g  e. 
_V
98elpw 3805 . . . . 5  |-  ( g  e.  ~P ( B  X.  A )  <->  g  C_  ( B  X.  A
) )
10 fneq1 5534 . . . . . 6  |-  ( f  =  g  ->  (
f  Fn  B  <->  g  Fn  B ) )
118, 10elab 3082 . . . . 5  |-  ( g  e.  { f  |  f  Fn  B }  <->  g  Fn  B )
129, 11anbi12i 679 . . . 4  |-  ( ( g  e.  ~P ( B  X.  A )  /\  g  e.  { f  |  f  Fn  B } )  <->  ( g  C_  ( B  X.  A
)  /\  g  Fn  B ) )
137, 12bitri 241 . . 3  |-  ( g  e.  ( ~P ( B  X.  A )  i^i 
{ f  |  f  Fn  B } )  <-> 
( g  C_  ( B  X.  A )  /\  g  Fn  B )
)
143, 6, 133bitr4i 269 . 2  |-  ( g  e.  ( A  ^m  B )  <->  g  e.  ( ~P ( B  X.  A )  i^i  {
f  |  f  Fn  B } ) )
1514eqriv 2433 1  |-  ( A  ^m  B )  =  ( ~P ( B  X.  A )  i^i 
{ f  |  f  Fn  B } )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   _Vcvv 2956    i^i cin 3319    C_ wss 3320   ~Pcpw 3799    X. cxp 4876    Fn wfn 5449   -->wf 5450  (class class class)co 6081    ^m cmap 7018
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020
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