Theorem dmarea 26914

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 25718	. . . 4 ⊢ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥 ∈ V
2		df-area 26913	. . . 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 6633	. . 3 ⊢ dom area = {𝑡 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿1)}
4	3	eleq2i 2825	. 2 ⊢ (𝐴 ∈ dom area ↔ 𝐴 ∈ {𝑡 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿1)})
5		imaeq1 6011	. . . . . 6 ⊢ (𝑡 = 𝐴 → (𝑡 “ {𝑥}) = (𝐴 “ {𝑥}))
6	5	eleq1d 2818	. . . . 5 ⊢ (𝑡 = 𝐴 → ((𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ↔ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ)))
7	6	ralbidv 3156	. . . 4 ⊢ (𝑡 = 𝐴 → (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ↔ ∀𝑥 ∈ ℝ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ)))
8	5	fveq2d 6835	. . . . . 6 ⊢ (𝑡 = 𝐴 → (vol‘(𝑡 “ {𝑥})) = (vol‘(𝐴 “ {𝑥})))
9	8	mpteq2dv 5189	. . . . 5 ⊢ (𝑡 = 𝐴 → (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) = (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))))
10	9	eleq1d 2818	. . . 4 ⊢ (𝑡 = 𝐴 → ((𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿1 ↔ (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))) ∈ 𝐿1))
11	7, 10	anbi12d 632	. . 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 3643	. 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 11108	. . . . . 6 ⊢ ℝ ∈ V
14	13, 13	xpex 7695	. . . . 5 ⊢ (ℝ × ℝ) ∈ V
15	14	elpw2 5276	. . . 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 278	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048  {crab 3396   ⊆ wss 3898  𝒫 cpw 4551  {csn 4577   ↦ cmpt 5176   × cxp 5619  ^◡ccnv 5620  dom cdm 5621   “ cima 5624  ‘cfv 6489  ℝcr 11016  volcvol 25411  𝐿¹cibl 25565  ∫citg 25566  areacarea 26912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-resscn 11074

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-fv 6497  df-sum 15601  df-itg 25571  df-area 26913

This theorem is referenced by:  areambl  26915  areass  26916  areaf  26918  areacirc  37826  arearect  43372  areaquad  43373