Theorem dmarea 26686

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 25512	. . . 4 ⊢ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥 ∈ V
2		df-area 26685	. . . 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 6694	. . 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 6054	. . . . . 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 3177	. . . 4 ⊢ (𝑡 = 𝐴 → (∀𝑥 ∈ ℝ (𝑡 “ {𝑥}) ∈ (◡vol “ ℝ) ↔ ∀𝑥 ∈ ℝ (𝐴 “ {𝑥}) ∈ (◡vol “ ℝ)))
8	5	fveq2d 6895	. . . . . 6 ⊢ (𝑡 = 𝐴 → (vol‘(𝑡 “ {𝑥})) = (vol‘(𝐴 “ {𝑥})))
9	8	mpteq2dv 5250	. . . . 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 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 3683	. 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 11203	. . . . . 6 ⊢ ℝ ∈ V
14	13, 13	xpex 7742	. . . . 5 ⊢ (ℝ × ℝ) ∈ V
15	14	elpw2 5345	. . . 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 1095	. . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  {crab 3432   ⊆ wss 3948  𝒫 cpw 4602  {csn 4628   ↦ cmpt 5231   × cxp 5674  ^◡ccnv 5675  dom cdm 5676   “ cima 5679  ‘cfv 6543  ℝcr 11111  volcvol 25204  𝐿¹cibl 25358  ∫citg 25359  areacarea 26684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-sum 15637  df-itg 25364  df-area 26685

This theorem is referenced by:  areambl  26687  areass  26688  areaf  26690  areacirc  36884  arearect  42266  areaquad  42267