Theorem omef 46463

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 46461	. . . . . 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 4622	. . . 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 2795	. . 3 ⊢ (𝜑 → 𝒫 𝑋 = dom 𝑂)
12	11	feq2d 6727	. 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 1538   ∈ wcel 2107  ∀wral 3060  ∅c0 4340  𝒫 cpw 4606  ∪ cuni 4913   class class class wbr 5149  dom cdm 5690   ↾ cres 5692  ⟶wf 6562  ‘cfv 6566  (class class class)co 7435  ωcom 7891   ≼ cdom 8988  0cc0 11159  +∞cpnf 11296   ≤ cle 11300  [,]cicc 13393  Σ^{^}csumge0 46329  OutMeascome 46456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3435  df-v 3481  df-dif 3967  df-un 3969  df-in 3971  df-ss 3981  df-nul 4341  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4914  df-br 5150  df-opab 5212  df-rel 5697  df-cnv 5698  df-co 5699  df-dm 5700  df-rn 5701  df-res 5702  df-iota 6519  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ome 46457

This theorem is referenced by:  omecl  46470  omeunle  46483  omeiunle  46484  caratheodory  46495