Theorem dfarea 26905

Description: Rewrite df-area 26901 self-referentially. (Contributed by Mario Carneiro, 21-Jun-2015.)

Assertion

Ref	Expression
dfarea	⊢ area = (𝑠 ∈ dom area ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)

Distinct variable group:   𝑥,𝑠

Proof of Theorem dfarea

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-area 26901	. 2 ⊢ area = (𝑠 ∈ {𝑦 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑦 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑦 “ {𝑥}))) ∈ 𝐿1)} ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)
2		itgex 25713	. . . 4 ⊢ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥 ∈ V
3	2, 1	dmmpti 6699	. . 3 ⊢ dom area = {𝑦 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑦 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑦 “ {𝑥}))) ∈ 𝐿1)}
4	3	mpteq1i 5244	. 2 ⊢ (𝑠 ∈ dom area ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥) = (𝑠 ∈ {𝑦 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑦 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑦 “ {𝑥}))) ∈ 𝐿1)} ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)
5	1, 4	eqtr4i 2759	1 ⊢ area = (𝑠 ∈ dom area ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)

Syntax hints:   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3058  {crab 3429  𝒫 cpw 4603  {csn 4629   ↦ cmpt 5231   × cxp 5676  ^◡ccnv 5677  dom cdm 5678   “ cima 5681  ‘cfv 6548  ℝcr 11138  volcvol 25405  𝐿¹cibl 25559  ∫citg 25560  areacarea 26900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fn 6551  df-sum 15666  df-itg 25565  df-area 26901

This theorem is referenced by:  areaf  26906  areaval  26909