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Theorem omedm 41650
Description: The domain of an outer measure is a power set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
omedm (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 dom 𝑂)

Proof of Theorem omedm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isome 41645 . . . 4 (𝑂 ∈ OutMeas → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂𝑦 ∈ 𝒫 𝑥(𝑂𝑦) ≤ (𝑂𝑥)) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂(𝑥 ≼ ω → (𝑂 𝑥) ≤ (Σ^‘(𝑂𝑥))))))
21ibi 259 . . 3 (𝑂 ∈ OutMeas → ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂𝑦 ∈ 𝒫 𝑥(𝑂𝑦) ≤ (𝑂𝑥)) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂(𝑥 ≼ ω → (𝑂 𝑥) ≤ (Σ^‘(𝑂𝑥)))))
32simplld 758 . 2 (𝑂 ∈ OutMeas → ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0))
43simplrd 760 1 (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 dom 𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  wral 3090  c0 4141  𝒫 cpw 4379   cuni 4673   class class class wbr 4888  dom cdm 5357  cres 5359  wf 6133  cfv 6137  (class class class)co 6924  ωcom 7345  cdom 8241  0cc0 10274  +∞cpnf 10410  cle 10414  [,]cicc 12494  Σ^csumge0 41513  OutMeascome 41640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-fv 6145  df-ome 41641
This theorem is referenced by:  caragenss  41655  omeunile  41656
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