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Theorem dfarea 23018
Description: Rewrite df-area 23014 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 23014 . 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 21912 . . . 4  |-  S. RR ( vol `  ( s
" { x }
) )  _d x  e.  _V
32, 1dmmpti 5708 . . 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 4527 . . 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 2499 1  |- area  =  ( s  e.  dom area  |->  S. RR ( vol `  ( s
" { x }
) )  _d x )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818   ~Pcpw 4010   {csn 4027    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998   dom cdm 4999   "cima 5002   ` cfv 5586   RRcr 9487   volcvol 21610   L^1cibl 21761   S.citg 21762  areacarea 23013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fn 5589  df-sum 13468  df-itg 21767  df-area 23014
This theorem is referenced by:  areaf  23019  areaval  23022
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