Theorem dfarea 25537

Description: Rewrite df-area 25533 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 25533	. 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 24370	. . . 4 ⊢ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥 ∈ V
3	2, 1	dmmpti 6491	. . 3 ⊢ dom area = {𝑦 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑦 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑦 “ {𝑥}))) ∈ 𝐿1)}
4	3	mpteq1i 5155	. 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 2847	1 ⊢ area = (𝑠 ∈ dom area ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)

Syntax hints:   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  {crab 3142  𝒫 cpw 4538  {csn 4566   ↦ cmpt 5145   × cxp 5552  ^◡ccnv 5553  dom cdm 5554   “ cima 5557  ‘cfv 6354  ℝcr 10535  volcvol 24063  𝐿¹cibl 24217  ∫citg 24218  areacarea 25532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fn 6357  df-sum 15042  df-itg 24223  df-area 25533

This theorem is referenced by:  areaf  25538  areaval  25541