Theorem omef 46776

Description: An outer measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
omef.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
omef.x	⊢ 𝑋 = ∪ dom 𝑂

Assertion

Ref	Expression
omef	⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞))

Proof of Theorem omef

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omef.o	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
2		isome 46774	. . . . . 6 ⊢ (𝑂 ∈ OutMeas → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))))
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦))))))
4	1, 3	mpbid 232	. . . 4 ⊢ (𝜑 → ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))))
5	4	simplld 768	. . 3 ⊢ (𝜑 → ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0))
6	5	simplld 768	. 2 ⊢ (𝜑 → 𝑂:dom 𝑂⟶(0[,]+∞))
7		omef.x	. . . . 5 ⊢ 𝑋 = ∪ dom 𝑂
8	7	pweqi 4571	. . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
9		simp-4r 784	. . . . 5 ⊢ (((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 ∪ dom 𝑂∀𝑧 ∈ 𝒫 𝑦(𝑂‘𝑧) ≤ (𝑂‘𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂‘∪ 𝑦) ≤ (Σ^‘(𝑂 ↾ 𝑦)))) → dom 𝑂 = 𝒫 ∪ dom 𝑂)
10	4, 9	syl 17	. . . 4 ⊢ (𝜑 → dom 𝑂 = 𝒫 ∪ dom 𝑂)
11	8, 10	eqtr4id 2791	. . 3 ⊢ (𝜑 → 𝒫 𝑋 = dom 𝑂)
12	11	feq2d 6647	. 2 ⊢ (𝜑 → (𝑂:𝒫 𝑋⟶(0[,]+∞) ↔ 𝑂:dom 𝑂⟶(0[,]+∞)))
13	6, 12	mpbird 257	1 ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∅c0 4286  𝒫 cpw 4555  ∪ cuni 4864   class class class wbr 5099  dom cdm 5625   ↾ cres 5627  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360  ωcom 7810   ≼ cdom 8885  0cc0 11030  +∞cpnf 11167   ≤ cle 11171  [,]cicc 13268  Σ^{^}csumge0 46642  OutMeascome 46769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ome 46770

This theorem is referenced by:  omecl  46783  omeunle  46796  omeiunle  46797  caratheodory  46808