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Theorem areass 26904
Description: A measurable region is a subset of ℝ × ℝ. (Contributed by Mario Carneiro, 21-Jun-2015.)
Assertion
Ref Expression
areass (𝑆 ∈ dom area → 𝑆 ⊆ (ℝ × ℝ))

Proof of Theorem areass
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dmarea 26902 . 2 (𝑆 ∈ dom area ↔ (𝑆 ⊆ (ℝ × ℝ) ∧ ∀𝑥 ∈ ℝ (𝑆 “ {𝑥}) ∈ (vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑆 “ {𝑥}))) ∈ 𝐿1))
21simp1bi 1143 1 (𝑆 ∈ dom area → 𝑆 ⊆ (ℝ × ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  wral 3058  wss 3947  {csn 4629  cmpt 5231   × cxp 5676  ccnv 5677  dom cdm 5678  cima 5681  cfv 6548  cr 11138  volcvol 25405  𝐿1cibl 25559  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-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196
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-pw 4605  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-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-fv 6556  df-sum 15666  df-itg 25565  df-area 26901
This theorem is referenced by: (None)
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