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Theorem dfarea 20255
Description: Rewrite df-area 20251 self-referentially. (Contributed by Mario Carneiro, 21-Jun-2015.)
Assertion
Ref Expression
dfarea  |- area  =  ( s  e.  dom area  |->  S. RR ( vol `  ( s
" { x }
) )  _d x )
Distinct variable group:    x, s

Proof of Theorem dfarea
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-area 20251 . 2  |- area  =  ( s  e.  { y  e.  ~P ( RR 
X.  RR )  |  ( A. x  e.  RR  ( y " { x } )  e.  ( `' vol " RR )  /\  (
x  e.  RR  |->  ( vol `  ( y
" { x }
) ) )  e.  L ^1 ) } 
|->  S. RR ( vol `  ( s " {
x } ) )  _d x )
2 itgex 19125 . . . 4  |-  S. RR ( vol `  ( s
" { x }
) )  _d x  e.  _V
32, 1dmmpti 5373 . . 3  |-  dom area  =  {
y  e.  ~P ( RR  X.  RR )  |  ( A. x  e.  RR  ( y " { x } )  e.  ( `' vol " RR )  /\  (
x  e.  RR  |->  ( vol `  ( y
" { x }
) ) )  e.  L ^1 ) }
4 mpteq1 4100 . . 3  |-  ( dom area  =  { y  e.  ~P ( RR  X.  RR )  |  ( A. x  e.  RR  (
y " { x } )  e.  ( `' vol " RR )  /\  ( x  e.  RR  |->  ( vol `  (
y " { x } ) ) )  e.  L ^1 ) }  ->  ( s  e.  dom area  |->  S. RR ( vol `  ( s
" { x }
) )  _d x )  =  ( s  e.  { y  e. 
~P ( RR  X.  RR )  |  ( A. x  e.  RR  ( y " {
x } )  e.  ( `' vol " RR )  /\  ( x  e.  RR  |->  ( vol `  (
y " { x } ) ) )  e.  L ^1 ) }  |->  S. RR ( vol `  ( s
" { x }
) )  _d x ) )
53, 4ax-mp 8 . 2  |-  ( s  e.  dom area  |->  S. RR ( vol `  ( s
" { x }
) )  _d x )  =  ( s  e.  { y  e. 
~P ( RR  X.  RR )  |  ( A. x  e.  RR  ( y " {
x } )  e.  ( `' vol " RR )  /\  ( x  e.  RR  |->  ( vol `  (
y " { x } ) ) )  e.  L ^1 ) }  |->  S. RR ( vol `  ( s
" { x }
) )  _d x )
61, 5eqtr4i 2306 1  |- area  =  ( s  e.  dom area  |->  S. RR ( vol `  ( s
" { x }
) )  _d x )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   ~Pcpw 3625   {csn 3640    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   dom cdm 4689   "cima 4692   ` cfv 5255   RRcr 8736   volcvol 18823   L ^1cibl 18972   S.citg 18973  areacarea 20250
This theorem is referenced by:  areaf  20256  areaval  20259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fn 5258  df-sum 12159  df-itg 18979  df-area 20251
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