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Theorem mapval2 6656
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 5640 . . . 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 6655 . . 3  |-  ( g  e.  ( A  ^m  B )  <->  g : B
--> A )
7 elin 3310 . . . 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 3573 . . . . 5  |-  ( g  e.  ~P ( B  X.  A )  <->  g  C_  ( B  X.  A
) )
9 vex 2733 . . . . . 6  |-  g  e. 
_V
10 fneq1 5286 . . . . . 6  |-  ( f  =  g  ->  (
f  Fn  B  <->  g  Fn  B ) )
119, 10elab 2874 . . . . 5  |-  ( g  e.  { f  |  f  Fn  B }  <->  g  Fn  B )
128, 11anbi12i 457 . . . 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 2167 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 1348    e. wcel 2141   {cab 2156   _Vcvv 2730    i^i cin 3120    C_ wss 3121   ~Pcpw 3566    X. cxp 4609    Fn wfn 5193   -->wf 5194  (class class class)co 5853    ^m cmap 6626
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-map 6628
This theorem is referenced by: (None)
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