Theorem omef 43576

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 43574	. . . . . 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 235	. . . 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 4506	. . . 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 2792	. . 3 ⊢ (𝜑 → 𝒫 𝑋 = dom 𝑂)
12	11	feq2d 6490	. 2 ⊢ (𝜑 → (𝑂:𝒫 𝑋⟶(0[,]+∞) ↔ 𝑂:dom 𝑂⟶(0[,]+∞)))
13	6, 12	mpbird 260	1 ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1542   ∈ wcel 2114  ∀wral 3053  ∅c0 4211  𝒫 cpw 4488  ∪ cuni 4796   class class class wbr 5030  dom cdm 5525   ↾ cres 5527  ⟶wf 6335  ‘cfv 6339  (class class class)co 7170  ωcom 7599   ≼ cdom 8553  0cc0 10615  +∞cpnf 10750   ≤ cle 10754  [,]cicc 12824  Σ^{^}csumge0 43442  OutMeascome 43569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rab 3062  df-v 3400  df-un 3848  df-in 3850  df-ss 3860  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-ome 43570

This theorem is referenced by:  omecl  43583  omeunle  43596  omeiunle  43597  caratheodory  43608