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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 (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 43574 . . . . . 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 235 . . . 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 768 . . 3 (𝜑 → ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0))
65simplld 768 . 2 (𝜑𝑂:dom 𝑂⟶(0[,]+∞))
7 omef.x . . . . 5 𝑋 = dom 𝑂
87pweqi 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 𝑂𝑧 ∈ 𝒫 𝑦(𝑂𝑧) ≤ (𝑂𝑦)) ∧ ∀𝑦 ∈ 𝒫 dom 𝑂(𝑦 ≼ ω → (𝑂 𝑦) ≤ (Σ^‘(𝑂𝑦)))) → dom 𝑂 = 𝒫 dom 𝑂)
104, 9syl 17 . . . 4 (𝜑 → dom 𝑂 = 𝒫 dom 𝑂)
118, 10eqtr4id 2792 . . 3 (𝜑 → 𝒫 𝑋 = dom 𝑂)
1211feq2d 6490 . 2 (𝜑 → (𝑂:𝒫 𝑋⟶(0[,]+∞) ↔ 𝑂:dom 𝑂⟶(0[,]+∞)))
136, 12mpbird 260 1 (𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))
Colors of variables: wff setvar class
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
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