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Theorem omef 39386
 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 (𝜑𝑂 ∈ OutMeas)
omef.x 𝑋 = dom 𝑂
Assertion
Ref Expression
omef (𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))

Proof of Theorem omef
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omef.o . . . . 5 (𝜑𝑂 ∈ OutMeas)
2 isome 39384 . . . . . 6 (𝑂 ∈ OutMeas → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))))
31, 2syl 17 . . . . 5 (𝜑 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦))))))
41, 3mpbid 221 . . . 4 (𝜑 → ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))))
54simplld 787 . . 3 (𝜑 → ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0))
65simplld 787 . 2 (𝜑𝑂:dom 𝑂⟶(0[,]+∞))
7 omef.x . . . . . 6 𝑋 = dom 𝑂
87pweqi 4112 . . . . 5 𝒫 𝑋 = 𝒫 dom 𝑂
98a1i 11 . . . 4 (𝜑 → 𝒫 𝑋 = 𝒫 dom 𝑂)
10 simp-4r 803 . . . . 5 (((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))) → dom 𝑂 = 𝒫 dom 𝑂)
114, 10syl 17 . . . 4 (𝜑 → dom 𝑂 = 𝒫 dom 𝑂)
129, 11eqtr4d 2647 . . 3 (𝜑 → 𝒫 𝑋 = dom 𝑂)
1312feq2d 5944 . 2 (𝜑 → (𝑂:𝒫 𝑋⟶(0[,]+∞) ↔ 𝑂:dom 𝑂⟶(0[,]+∞)))
146, 13mpbird 246 1 (𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583  dom cdm 5038   ↾ cres 5040  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ωcom 6957   ≼ cdom 7839  0cc0 9815  +∞cpnf 9950   ≤ cle 9954  [,]cicc 12049  Σ^csumge0 39255  OutMeascome 39379 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ome 39380 This theorem is referenced by:  omecl  39393  omeunle  39406  omeiunle  39407  caratheodory  39418
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