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Theorem volres 21674
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 5496 . 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 21612 . . . 4  |-  vol  =  ( vol*  |`  { x  |  A. y  e.  ( `' vol* " RR ) ( vol* `  y )  =  ( ( vol* `  ( y  i^i  x
) )  +  ( vol* `  (
y  \  x )
) ) } )
32dmeqi 5202 . . 3  |-  dom  vol  =  dom  ( vol*  |` 
{ x  |  A. y  e.  ( `' vol* " RR ) ( vol* `  y )  =  ( ( vol* `  ( y  i^i  x
) )  +  ( vol* `  (
y  \  x )
) ) } )
43reseq2i 5268 . 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 2507 1  |-  vol  =  ( vol*  |`  dom  vol )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   {cab 2452   A.wral 2814    \ cdif 3473    i^i cin 3475   `'ccnv 4998   dom cdm 4999    |` cres 5001   "cima 5002   ` cfv 5586  (class class class)co 6282   RRcr 9487    + caddc 9491   vol*covol 21609   volcvol 21610
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-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-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-vol 21612
This theorem is referenced by:  volf  21675  mblvol  21676
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