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Theorem dfarea 23491
Description: Rewrite df-area 23487 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 23487 . 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 22346 . . . 4  |-  S. RR ( vol `  ( s
" { x }
) )  _d x  e.  _V
32, 1dmmpti 5692 . . 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 4519 . . 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 5 . 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 2486 1  |- area  =  ( s  e.  dom area  |->  S. RR ( vol `  ( s
" { x }
) )  _d x )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808   ~Pcpw 3999   {csn 4016    |-> cmpt 4497    X. cxp 4986   `'ccnv 4987   dom cdm 4988   "cima 4991   ` cfv 5570   RRcr 9480   volcvol 22044   L^1cibl 22195   S.citg 22196  areacarea 23486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fn 5573  df-sum 13594  df-itg 22201  df-area 23487
This theorem is referenced by:  areaf  23492  areaval  23495
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