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Theorem omef 44811
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 44809 . . . . . 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 231 . . . 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 767 . . 3 (𝜑 → ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0))
65simplld 767 . 2 (𝜑𝑂:dom 𝑂⟶(0[,]+∞))
7 omef.x . . . . 5 𝑋 = dom 𝑂
87pweqi 4581 . . . 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 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 2796 . . 3 (𝜑 → 𝒫 𝑋 = dom 𝑂)
1211feq2d 6659 . 2 (𝜑 → (𝑂:𝒫 𝑋⟶(0[,]+∞) ↔ 𝑂:dom 𝑂⟶(0[,]+∞)))
136, 12mpbird 257 1 (𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3065  c0 4287  𝒫 cpw 4565   cuni 4870   class class class wbr 5110  dom cdm 5638  cres 5640  wf 6497  cfv 6501  (class class class)co 7362  ωcom 7807  cdom 8888  0cc0 11058  +∞cpnf 11193  cle 11197  [,]cicc 13274  Σ^csumge0 44677  OutMeascome 44804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ome 44805
This theorem is referenced by:  omecl  44818  omeunle  44831  omeiunle  44832  caratheodory  44843
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