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Theorem dfarea 22239
Description: Rewrite df-area 22235 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 22235 . 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 21090 . . . 4  |-  S. RR ( vol `  ( s
" { x }
) )  _d x  e.  _V
32, 1dmmpti 5528 . . 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 4360 . . 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 2456 1  |- area  =  ( s  e.  dom area  |->  S. RR ( vol `  ( s
" { x }
) )  _d x )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1362    e. wcel 1755   A.wral 2705   {crab 2709   ~Pcpw 3848   {csn 3865    e. cmpt 4338    X. cxp 4825   `'ccnv 4826   dom cdm 4827   "cima 4830   ` cfv 5406   RRcr 9269   volcvol 20789   L^1cibl 20939   S.citg 20940  areacarea 22234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-iota 5369  df-fun 5408  df-fn 5409  df-sum 13148  df-itg 20945  df-area 22235
This theorem is referenced by:  areaf  22240  areaval  22243
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