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Theorem areass 26842
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 26840 . 2 (𝑆 ∈ dom area ↔ (𝑆 ⊆ (ℝ × ℝ) ∧ ∀𝑥 ∈ ℝ (𝑆 “ {𝑥}) ∈ (vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑆 “ {𝑥}))) ∈ 𝐿1))
21simp1bi 1142 1 (𝑆 ∈ dom area → 𝑆 ⊆ (ℝ × ℝ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wral 3055  wss 3943  {csn 4623  cmpt 5224   × cxp 5667  ccnv 5668  dom cdm 5669  cima 5672  cfv 6536  cr 11108  volcvol 25343  𝐿1cibl 25497  areacarea 26838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-fv 6544  df-sum 15637  df-itg 25503  df-area 26839
This theorem is referenced by: (None)
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