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Theorem volres 23496
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 5786 . 2 (vol* ↾ dom (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦𝑥)) + (vol*‘(𝑦𝑥)))})) = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦𝑥)) + (vol*‘(𝑦𝑥)))})
2 df-vol 23434 . . . 4 vol = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦𝑥)) + (vol*‘(𝑦𝑥)))})
32dmeqi 5480 . . 3 dom vol = dom (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦𝑥)) + (vol*‘(𝑦𝑥)))})
43reseq2i 5548 . 2 (vol* ↾ dom vol) = (vol* ↾ dom (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦𝑥)) + (vol*‘(𝑦𝑥)))}))
51, 4, 23eqtr4ri 2793 1 vol = (vol* ↾ dom vol)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  {cab 2746  wral 3050  cdif 3712  cin 3714  ccnv 5265  dom cdm 5266  cres 5268  cima 5269  cfv 6049  (class class class)co 6813  cr 10127   + caddc 10131  vol*covol 23431  volcvol 23432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-vol 23434
This theorem is referenced by:  volf  23497  mblvol  23498
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