Theorem dmarea 26107

Description: The domain of the area function is the set of finitely measurable subsets of ℝ × ℝ. (Contributed by Mario Carneiro, 21-Jun-2015.)

Assertion

Ref	Expression
dmarea	⊢ (𝐴 ∈ dom area ↔ (𝐴 ⊆ (ℝ × ℝ) ∧ ∀𝑥 ∈ ℝ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))) ∈ 𝐿1))

Distinct variable group:   𝑥,𝐴

Proof of Theorem dmarea

Dummy variables 𝑡 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		itgex 24935	. . . 4 ⊢ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥 ∈ V
2		df-area 26106	. . . 4 ⊢ area = (𝑠 ∈ {𝑡 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿1)} ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)
3	1, 2	dmmpti 6577	. . 3 ⊢ dom area = {𝑡 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿1)}
4	3	eleq2i 2830	. 2 ⊢ (𝐴 ∈ dom area ↔ 𝐴 ∈ {𝑡 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿1)})
5		imaeq1 5964	. . . . . 6 ⊢ (𝑡 = 𝐴 → (𝑡 “ {𝑥}) = (𝐴 “ {𝑥}))
6	5	eleq1d 2823	. . . . 5 ⊢ (𝑡 = 𝐴 → ((𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ↔ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ)))
7	6	ralbidv 3112	. . . 4 ⊢ (𝑡 = 𝐴 → (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ↔ ∀𝑥 ∈ ℝ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ)))
8	5	fveq2d 6778	. . . . . 6 ⊢ (𝑡 = 𝐴 → (vol‘(𝑡 “ {𝑥})) = (vol‘(𝐴 “ {𝑥})))
9	8	mpteq2dv 5176	. . . . 5 ⊢ (𝑡 = 𝐴 → (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) = (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))))
10	9	eleq1d 2823	. . . 4 ⊢ (𝑡 = 𝐴 → ((𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿1 ↔ (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))) ∈ 𝐿1))
11	7, 10	anbi12d 631	. . 3 ⊢ (𝑡 = 𝐴 → ((∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿1) ↔ (∀𝑥 ∈ ℝ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))) ∈ 𝐿1)))
12	11	elrab 3624	. 2 ⊢ (𝐴 ∈ {𝑡 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿1)} ↔ (𝐴 ∈ 𝒫 (ℝ × ℝ) ∧ (∀𝑥 ∈ ℝ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))) ∈ 𝐿1)))
13		reex 10962	. . . . . 6 ⊢ ℝ ∈ V
14	13, 13	xpex 7603	. . . . 5 ⊢ (ℝ × ℝ) ∈ V
15	14	elpw2 5269	. . . 4 ⊢ (𝐴 ∈ 𝒫 (ℝ × ℝ) ↔ 𝐴 ⊆ (ℝ × ℝ))
16	15	anbi1i 624	. . 3 ⊢ ((𝐴 ∈ 𝒫 (ℝ × ℝ) ∧ (∀𝑥 ∈ ℝ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))) ∈ 𝐿1)) ↔ (𝐴 ⊆ (ℝ × ℝ) ∧ (∀𝑥 ∈ ℝ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))) ∈ 𝐿1)))
17		3anass 1094	. . 3 ⊢ ((𝐴 ⊆ (ℝ × ℝ) ∧ ∀𝑥 ∈ ℝ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))) ∈ 𝐿1) ↔ (𝐴 ⊆ (ℝ × ℝ) ∧ (∀𝑥 ∈ ℝ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))) ∈ 𝐿1)))
18	16, 17	bitr4i 277	. 2 ⊢ ((𝐴 ∈ 𝒫 (ℝ × ℝ) ∧ (∀𝑥 ∈ ℝ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))) ∈ 𝐿1)) ↔ (𝐴 ⊆ (ℝ × ℝ) ∧ ∀𝑥 ∈ ℝ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))) ∈ 𝐿1))
19	4, 12, 18	3bitri 297	1 ⊢ (𝐴 ∈ dom area ↔ (𝐴 ⊆ (ℝ × ℝ) ∧ ∀𝑥 ∈ ℝ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))) ∈ 𝐿1))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068   ⊆ wss 3887  𝒫 cpw 4533  {csn 4561   ↦ cmpt 5157   × cxp 5587  ^◡ccnv 5588  dom cdm 5589   “ cima 5592  ‘cfv 6433  ℝcr 10870  volcvol 24627  𝐿¹cibl 24781  ∫citg 24782  areacarea 26105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-sum 15398  df-itg 24787  df-area 26106

This theorem is referenced by:  areambl  26108  areass  26109  areaf  26111  areacirc  35870  arearect  41046  areaquad  41047