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Theorem mblvol 23638
Description: The volume of a measurable set is the same as its outer volume. (Contributed by Mario Carneiro, 17-Mar-2014.)
Assertion
Ref Expression
mblvol (𝐴 ∈ dom vol → (vol‘𝐴) = (vol*‘𝐴))

Proof of Theorem mblvol
StepHypRef Expression
1 volres 23636 . . 3 vol = (vol* ↾ dom vol)
21fveq1i 6412 . 2 (vol‘𝐴) = ((vol* ↾ dom vol)‘𝐴)
3 fvres 6430 . 2 (𝐴 ∈ dom vol → ((vol* ↾ dom vol)‘𝐴) = (vol*‘𝐴))
42, 3syl5eq 2845 1 (𝐴 ∈ dom vol → (vol‘𝐴) = (vol*‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  dom cdm 5312  cres 5314  cfv 6101  vol*covol 23570  volcvol 23571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320  df-dm 5322  df-rn 5323  df-res 5324  df-iota 6064  df-fv 6109  df-vol 23573
This theorem is referenced by:  volss  23641  volun  23653  volinun  23654  volfiniun  23655  voliunlem3  23660  volsup  23664  iccvolcl  23675  ovolioo  23676  volioo  23677  ioovolcl  23678  uniioovol  23687  uniioombllem4  23694  volcn  23714  volivth  23715  vitalilem4  23719  i1fima2  23787  i1fd  23789  i1f0rn  23790  itg1val2  23792  itg1ge0  23794  itg11  23799  i1fadd  23803  i1fmul  23804  itg1addlem2  23805  itg1addlem4  23807  i1fres  23813  itg10a  23818  itg1ge0a  23819  itg1climres  23822  mbfi1fseqlem4  23826  itg2const2  23849  itg2gt0  23868  itg2cnlem2  23870  ftc1a  24141  ftc1lem4  24143  itgulm  24503  areaf  25040  cntnevol  30807  volmeas  30810  mblfinlem3  33937  mblfinlem4  33938  ismblfin  33939  voliunnfl  33942  volsupnfl  33943  itg2addnclem  33949  itg2addnclem2  33950  itg2gt0cn  33953  ftc1cnnclem  33971  ftc1anclem7  33979  areacirc  33993  arearect  38581  areaquad  38582  vol0  40914  volge0  40916  volsn  40922  volicc  40954  vonvol  41618
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