Theorem omef 46419

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 46417	. . . . . 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 767	. . 3 ⊢ (𝜑 → ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 ∪ dom 𝑂) ∧ (𝑂‘∅) = 0))
6	5	simplld 767	. 2 ⊢ (𝜑 → 𝑂:dom 𝑂⟶(0[,]+∞))
7		omef.x	. . . . 5 ⊢ 𝑋 = ∪ dom 𝑂
8	7	pweqi 4638	. . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
9		simp-4r 783	. . . . 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 2799	. . 3 ⊢ (𝜑 → 𝒫 𝑋 = dom 𝑂)
12	11	feq2d 6735	. 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 1537   ∈ wcel 2108  ∀wral 3067  ∅c0 4352  𝒫 cpw 4622  ∪ cuni 4931   class class class wbr 5166  dom cdm 5700   ↾ cres 5702  ⟶wf 6571  ‘cfv 6575  (class class class)co 7450  ωcom 7905   ≼ cdom 9003  0cc0 11186  +∞cpnf 11323   ≤ cle 11327  [,]cicc 13412  Σ^{^}csumge0 46285  OutMeascome 46412

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-fv 6583  df-ome 46413

This theorem is referenced by:  omecl  46426  omeunle  46439  omeiunle  46440  caratheodory  46451