Theorem omedm 46514

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.

Step	Hyp	Ref	Expression
1		isome 46509	. . . 4 ⊢ (𝑂 ∈ OutMeas → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 ∪ dom 𝑂∀𝑦 ∈ 𝒫 𝑥(𝑂‘𝑦) ≤ (𝑂‘𝑥)) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂(𝑥 ≼ ω → (𝑂‘∪ 𝑥) ≤ (Σ^‘(𝑂 ↾ 𝑥))))))
2	1	ibi 267	. . 3 ⊢ (𝑂 ∈ OutMeas → ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 ∪ dom 𝑂∀𝑦 ∈ 𝒫 𝑥(𝑂‘𝑦) ≤ (𝑂‘𝑥)) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂(𝑥 ≼ ω → (𝑂‘∪ 𝑥) ≤ (Σ^‘(𝑂 ↾ 𝑥)))))
3	2	simplld 768	. 2 ⊢ (𝑂 ∈ OutMeas → ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0))
4	3	simplrd 770	1 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 ∪ dom 𝑂)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∅c0 4333  𝒫 cpw 4600  ∪ cuni 4907   class class class wbr 5143  dom cdm 5685   ↾ cres 5687  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  ωcom 7887   ≼ cdom 8983  0cc0 11155  +∞cpnf 11292   ≤ cle 11296  [,]cicc 13390  Σ^{^}csumge0 46377  OutMeascome 46504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ome 46505

This theorem is referenced by:  caragenss  46519  omeunile  46520