Theorem volres 25495

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.

Step	Hyp	Ref	Expression
1		resdmres 6196	. 2 ⊢ (vol* ↾ dom (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))})) = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))})
2		df-vol 25432	. . . 4 ⊢ vol = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))})
3	2	dmeqi 5859	. . 3 ⊢ dom vol = dom (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))})
4	3	reseq2i 5941	. 2 ⊢ (vol* ↾ dom vol) = (vol* ↾ dom (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))}))
5	1, 4, 2	3eqtr4ri 2770	1 ⊢ vol = (vol* ↾ dom vol)

Syntax hints:   = wceq 1542  {cab 2714  ∀wral 3051   ∖ cdif 3886   ∩ cin 3888  ^◡ccnv 5630  dom cdm 5631   ↾ cres 5633   “ cima 5634  ‘cfv 6498  (class class class)co 7367  ℝcr 11037   + caddc 11041  vol*covol 25429  volcvol 25430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-vol 25432

This theorem is referenced by:  volf  25496  mblvol  25497