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Theorem volres 22233
Description: A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.)
Assertion
Ref Expression
volres  |-  vol  =  ( vol*  |`  dom  vol )

Proof of Theorem volres
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resdmres 5316 . 2  |-  ( vol*  |`  dom  ( vol*  |`  { x  | 
A. y  e.  ( `' vol* " RR ) ( vol* `  y )  =  ( ( vol* `  ( y  i^i  x
) )  +  ( vol* `  (
y  \  x )
) ) } ) )  =  ( vol*  |`  { x  | 
A. y  e.  ( `' vol* " RR ) ( vol* `  y )  =  ( ( vol* `  ( y  i^i  x
) )  +  ( vol* `  (
y  \  x )
) ) } )
2 df-vol 22171 . . . 4  |-  vol  =  ( vol*  |`  { x  |  A. y  e.  ( `' vol* " RR ) ( vol* `  y )  =  ( ( vol* `  ( y  i^i  x
) )  +  ( vol* `  (
y  \  x )
) ) } )
32dmeqi 5027 . . 3  |-  dom  vol  =  dom  ( vol*  |` 
{ x  |  A. y  e.  ( `' vol* " RR ) ( vol* `  y )  =  ( ( vol* `  ( y  i^i  x
) )  +  ( vol* `  (
y  \  x )
) ) } )
43reseq2i 5093 . 2  |-  ( vol*  |`  dom  vol )  =  ( vol*  |` 
dom  ( vol*  |` 
{ x  |  A. y  e.  ( `' vol* " RR ) ( vol* `  y )  =  ( ( vol* `  ( y  i^i  x
) )  +  ( vol* `  (
y  \  x )
) ) } ) )
51, 4, 23eqtr4ri 2444 1  |-  vol  =  ( vol*  |`  dom  vol )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1407   {cab 2389   A.wral 2756    \ cdif 3413    i^i cin 3415   `'ccnv 4824   dom cdm 4825    |` cres 4827   "cima 4828   ` cfv 5571  (class class class)co 6280   RRcr 9523    + caddc 9527   vol*covol 22168   volcvol 22169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-xp 4831  df-rel 4832  df-cnv 4833  df-dm 4835  df-rn 4836  df-res 4837  df-vol 22171
This theorem is referenced by:  volf  22234  mblvol  22235
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