Theorem dfarea 27021

Description: Rewrite df-area 27017 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 27017	. 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 25825	. . . 4 ⊢ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥 ∈ V
3	2, 1	dmmpti 6724	. . 3 ⊢ dom area = {𝑦 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑦 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑦 “ {𝑥}))) ∈ 𝐿1)}
4	3	mpteq1i 5262	. 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 2771	1 ⊢ area = (𝑠 ∈ dom area ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)

Syntax hints:   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443  𝒫 cpw 4622  {csn 4648   ↦ cmpt 5249   × cxp 5698  ^◡ccnv 5699  dom cdm 5700   “ cima 5703  ‘cfv 6573  ℝcr 11183  volcvol 25517  𝐿¹cibl 25671  ∫citg 25672  areacarea 27016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fn 6576  df-sum 15735  df-itg 25677  df-area 27017

This theorem is referenced by:  areaf  27022  areaval  27025