Theorem dmarea 26012

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 24840	. . . 4 ⊢ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥 ∈ V
2		df-area 26011	. . . 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 6561	. . 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 5953	. . . . . 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 3120	. . . 4 ⊢ (𝑡 = 𝐴 → (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ↔ ∀𝑥 ∈ ℝ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ)))
8	5	fveq2d 6760	. . . . . 6 ⊢ (𝑡 = 𝐴 → (vol‘(𝑡 “ {𝑥})) = (vol‘(𝐴 “ {𝑥})))
9	8	mpteq2dv 5172	. . . . 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 630	. . 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 3617	. 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 10893	. . . . . 6 ⊢ ℝ ∈ V
14	13, 13	xpex 7581	. . . . 5 ⊢ (ℝ × ℝ) ∈ V
15	14	elpw2 5264	. . . 4 ⊢ (𝐴 ∈ 𝒫 (ℝ × ℝ) ↔ 𝐴 ⊆ (ℝ × ℝ))
16	15	anbi1i 623	. . 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 1093	. . 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 296	1 ⊢ (𝐴 ∈ dom area ↔ (𝐴 ⊆ (ℝ × ℝ) ∧ ∀𝑥 ∈ ℝ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))) ∈ 𝐿1))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067   ⊆ wss 3883  𝒫 cpw 4530  {csn 4558   ↦ cmpt 5153   × cxp 5578  ^◡ccnv 5579  dom cdm 5580   “ cima 5583  ‘cfv 6418  ℝcr 10801  volcvol 24532  𝐿¹cibl 24686  ∫citg 24687  areacarea 26010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426  df-sum 15326  df-itg 24692  df-area 26011

This theorem is referenced by:  areambl  26013  areass  26014  areaf  26016  areacirc  35797  arearect  40962  areaquad  40963