MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfarea Structured version   Unicode version

Theorem dfarea 23873
Description: Rewrite df-area 23869 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 23869 . 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 22715 . . . 4  |-  S. RR ( vol `  ( s
" { x }
) )  _d x  e.  _V
32, 1dmmpti 5722 . . 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 4501 . . 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 2454 1  |- area  =  ( s  e.  dom area  |->  S. RR ( vol `  ( s
" { x }
) )  _d x )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437    e. wcel 1868   A.wral 2775   {crab 2779   ~Pcpw 3979   {csn 3996    |-> cmpt 4479    X. cxp 4848   `'ccnv 4849   dom cdm 4850   "cima 4853   ` cfv 5598   RRcr 9539   volcvol 22402   L^1cibl 22562   S.citg 22563  areacarea 23868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-iota 5562  df-fun 5600  df-fn 5601  df-sum 13741  df-itg 22568  df-area 23869
This theorem is referenced by:  areaf  23874  areaval  23877
  Copyright terms: Public domain W3C validator