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Theorem areass 24486
 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 24484 . 2 (𝑆 ∈ dom area ↔ (𝑆 ⊆ (ℝ × ℝ) ∧ ∀𝑥 ∈ ℝ (𝑆 “ {𝑥}) ∈ (vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑆 “ {𝑥}))) ∈ 𝐿1))
21simp1bi 1069 1 (𝑆 ∈ dom area → 𝑆 ⊆ (ℝ × ℝ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  {csn 4125   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037  dom cdm 5038   “ cima 5041  ‘cfv 5804  ℝcr 9814  volcvol 23039  𝐿1cibl 23192  areacarea 24482 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-sum 14265  df-itg 23198  df-area 24483 This theorem is referenced by: (None)
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