Theorem omedm 47453

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 47448	. . . 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 270	. . 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 780	. 2 ⊢ (𝑂 ∈ OutMeas → ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0))
4	3	simplrd 782	1 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 ∪ dom 𝑂)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∅c0 4279  𝒫 cpw 4557  ∪ cuni 4867   class class class wbr 5103  dom cdm 5651   ↾ cres 5653  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412  ωcom 7866   ≼ cdom 8955  0cc0 11181  +∞cpnf 11321   ≤ cle 11325  [,]cicc 13460  Σ^{^}csumge0 47316  OutMeascome 47443

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ome 47444

This theorem is used by:  caragenss  47458  omeunile  47459