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Mirrors > Home > MPE Home > Th. List > volres | Structured version Visualization version GIF version |
Description: A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.) |
Ref | Expression |
---|---|
volres | ⊢ vol = (vol* ↾ dom vol) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resdmres 5543 | . 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 23041 | . . . 4 ⊢ vol = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))}) | |
3 | 2 | dmeqi 5247 | . . 3 ⊢ dom vol = dom (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))}) |
4 | 3 | reseq2i 5314 | . 2 ⊢ (vol* ↾ dom vol) = (vol* ↾ dom (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))})) |
5 | 1, 4, 2 | 3eqtr4ri 2643 | 1 ⊢ vol = (vol* ↾ dom vol) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 {cab 2596 ∀wral 2896 ∖ cdif 3537 ∩ cin 3539 ◡ccnv 5037 dom cdm 5038 ↾ cres 5040 “ cima 5041 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 + caddc 9818 vol*covol 23038 volcvol 23039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-vol 23041 |
This theorem is referenced by: volf 23104 mblvol 23105 |
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