Theorem dmarea 24684

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 23537	. . . 4 ⊢ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥 ∈ V
2		df-area 24683	. . . 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 6023	. . 3 ⊢ dom area = {𝑡 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿1)}
4	3	eleq2i 2693	. 2 ⊢ (𝐴 ∈ dom area ↔ 𝐴 ∈ {𝑡 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿1)})
5		imaeq1 5461	. . . . . 6 ⊢ (𝑡 = 𝐴 → (𝑡 “ {𝑥}) = (𝐴 “ {𝑥}))
6	5	eleq1d 2686	. . . . 5 ⊢ (𝑡 = 𝐴 → ((𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ↔ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ)))
7	6	ralbidv 2986	. . . 4 ⊢ (𝑡 = 𝐴 → (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ↔ ∀𝑥 ∈ ℝ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ)))
8	5	fveq2d 6195	. . . . . 6 ⊢ (𝑡 = 𝐴 → (vol‘(𝑡 “ {𝑥})) = (vol‘(𝐴 “ {𝑥})))
9	8	mpteq2dv 4745	. . . . 5 ⊢ (𝑡 = 𝐴 → (𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) = (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))))
10	9	eleq1d 2686	. . . 4 ⊢ (𝑡 = 𝐴 → ((𝑥 ∈ ℝ ↦ (vol‘(𝑡 “ {𝑥}))) ∈ 𝐿1 ↔ (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))) ∈ 𝐿1))
11	7, 10	anbi12d 747	. . 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 3363	. 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 10027	. . . . . 6 ⊢ ℝ ∈ V
14	13, 13	xpex 6962	. . . . 5 ⊢ (ℝ × ℝ) ∈ V
15	14	elpw2 4828	. . . 4 ⊢ (𝐴 ∈ 𝒫 (ℝ × ℝ) ↔ 𝐴 ⊆ (ℝ × ℝ))
16	15	anbi1i 731	. . 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 1042	. . 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 267	. 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 286	1 ⊢ (𝐴 ∈ dom area ↔ (𝐴 ⊆ (ℝ × ℝ) ∧ ∀𝑥 ∈ ℝ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝐴 “ {𝑥}))) ∈ 𝐿1))

Syntax hints:   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916   ⊆ wss 3574  𝒫 cpw 4158  {csn 4177   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113  dom cdm 5114   “ cima 5117  ‘cfv 5888  ℝcr 9935  volcvol 23232  𝐿¹cibl 23386  ∫citg 23387  areacarea 24682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-sum 14417  df-itg 23392  df-area 24683

This theorem is referenced by:  areambl  24685  areass  24686  areaf  24688  areacirc  33505  arearect  37801  areaquad  37802