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Theorem mapval2 6624
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 5612 . . . 4  |-  ( g : B --> A  <->  ( g  Fn  B  /\  g  C_  ( B  X.  A
) ) )
2 ancom 264 . . . 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 6623 . . 3  |-  ( g  e.  ( A  ^m  B )  <->  g : B
--> A )
7 elin 3290 . . . 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 velpw 3550 . . . . 5  |-  ( g  e.  ~P ( B  X.  A )  <->  g  C_  ( B  X.  A
) )
9 vex 2715 . . . . . 6  |-  g  e. 
_V
10 fneq1 5259 . . . . . 6  |-  ( f  =  g  ->  (
f  Fn  B  <->  g  Fn  B ) )
119, 10elab 2856 . . . . 5  |-  ( g  e.  { f  |  f  Fn  B }  <->  g  Fn  B )
128, 11anbi12i 456 . . . 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 2154 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 1335    e. wcel 2128   {cab 2143   _Vcvv 2712    i^i cin 3101    C_ wss 3102   ~Pcpw 3543    X. cxp 4585    Fn wfn 5166   -->wf 5167  (class class class)co 5825    ^m cmap 6594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-fv 5179  df-ov 5828  df-oprab 5829  df-mpo 5830  df-map 6596
This theorem is referenced by: (None)
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