MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfarea Structured version   Visualization version   GIF version

Theorem dfarea 24908
Description: Rewrite df-area 24904 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.
StepHypRef Expression
1 df-area 24904 . 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 23757 . . . 4 ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥 ∈ V
32, 1dmmpti 6162 . . 3 dom area = {𝑦 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑦 “ {𝑥}) ∈ (vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑦 “ {𝑥}))) ∈ 𝐿1)}
4 mpteq1 4872 . . 3 (dom area = {𝑦 ∈ 𝒫 (ℝ × ℝ) ∣ (∀𝑥 ∈ ℝ (𝑦 “ {𝑥}) ∈ (vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑦 “ {𝑥}))) ∈ 𝐿1)} → (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))
53, 4ax-mp 5 . 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)
61, 5eqtr4i 2796 1 area = (𝑠 ∈ dom area ↦ ∫ℝ(vol‘(𝑠 “ {𝑥})) d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  𝒫 cpw 4298  {csn 4317  cmpt 4864   × cxp 5248  ccnv 5249  dom cdm 5250  cima 5253  cfv 6030  cr 10141  volcvol 23451  𝐿1cibl 23605  citg 23606  areacarea 24903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fn 6033  df-sum 14625  df-itg 23611  df-area 24904
This theorem is referenced by:  areaf  24909  areaval  24912
  Copyright terms: Public domain W3C validator