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Mirrors > Home > MPE Home > Th. List > dfarea | Structured version Visualization version GIF version |
Description: Rewrite df-area 27014 self-referentially. (Contributed by Mario Carneiro, 21-Jun-2015.) |
Ref | Expression |
---|---|
dfarea | ⊢ area = (𝑠 ∈ dom area ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-area 27014 | . 2 ⊢ area = (𝑠 ∈ {𝑦 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑦 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑦 “ {𝑥}))) ∈ 𝐿1)} ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥) | |
2 | itgex 25820 | . . . 4 ⊢ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥 ∈ V | |
3 | 2, 1 | dmmpti 6713 | . . 3 ⊢ dom area = {𝑦 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑦 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑦 “ {𝑥}))) ∈ 𝐿1)} |
4 | 3 | mpteq1i 5244 | . 2 ⊢ (𝑠 ∈ dom area ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥) = (𝑠 ∈ {𝑦 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑦 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑦 “ {𝑥}))) ∈ 𝐿1)} ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥) |
5 | 1, 4 | eqtr4i 2766 | 1 ⊢ area = (𝑠 ∈ dom area ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 𝒫 cpw 4605 {csn 4631 ↦ cmpt 5231 × cxp 5687 ◡ccnv 5688 dom cdm 5689 “ cima 5692 ‘cfv 6563 ℝcr 11152 volcvol 25512 𝐿1cibl 25666 ∫citg 25667 areacarea 27013 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fn 6566 df-sum 15720 df-itg 25672 df-area 27014 |
This theorem is referenced by: areaf 27019 areaval 27022 |
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