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Theorem omedm 39389
Description: The domain of an outer measure is a power set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
omedm (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 dom 𝑂)

Proof of Theorem omedm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4 (𝑂 ∈ OutMeas → 𝑂 ∈ OutMeas)
2 isome 39384 . . . 4 (𝑂 ∈ OutMeas → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂𝑦 ∈ 𝒫 𝑥(𝑂𝑦) ≤ (𝑂𝑥)) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂(𝑥 ≼ ω → (𝑂 𝑥) ≤ (Σ^‘(𝑂𝑥))))))
31, 2mpbid 221 . . 3 (𝑂 ∈ OutMeas → ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂𝑦 ∈ 𝒫 𝑥(𝑂𝑦) ≤ (𝑂𝑥)) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂(𝑥 ≼ ω → (𝑂 𝑥) ≤ (Σ^‘(𝑂𝑥)))))
43simplld 787 . 2 (𝑂 ∈ OutMeas → ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0))
54simplrd 789 1 (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 dom 𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  c0 3874  𝒫 cpw 4108   cuni 4372   class class class wbr 4583  dom cdm 5038  cres 5040  wf 5800  cfv 5804  (class class class)co 6549  ωcom 6957  cdom 7839  0cc0 9815  +∞cpnf 9950  cle 9954  [,]cicc 12049  Σ^csumge0 39255  OutMeascome 39379
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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  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-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ome 39380
This theorem is referenced by:  caragenss  39394  omeunile  39395
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