Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  omecl Structured version   Visualization version   GIF version

Theorem omecl 43081
Description: The outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
omecl.o (𝜑𝑂 ∈ OutMeas)
omecl.x 𝑋 = dom 𝑂
omecl.ss (𝜑𝐴𝑋)
Assertion
Ref Expression
omecl (𝜑 → (𝑂𝐴) ∈ (0[,]+∞))

Proof of Theorem omecl
StepHypRef Expression
1 omecl.o . . 3 (𝜑𝑂 ∈ OutMeas)
2 omecl.x . . 3 𝑋 = dom 𝑂
31, 2omef 43074 . 2 (𝜑𝑂:𝒫 𝑋⟶(0[,]+∞))
4 omecl.ss . . 3 (𝜑𝐴𝑋)
52a1i 11 . . . . . 6 (𝜑𝑋 = dom 𝑂)
61dmexd 7601 . . . . . . 7 (𝜑 → dom 𝑂 ∈ V)
76uniexd 7453 . . . . . 6 (𝜑 dom 𝑂 ∈ V)
85, 7eqeltrd 2914 . . . . 5 (𝜑𝑋 ∈ V)
98, 4ssexd 5204 . . . 4 (𝜑𝐴 ∈ V)
10 elpwg 4514 . . . 4 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
119, 10syl 17 . . 3 (𝜑 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
124, 11mpbird 260 . 2 (𝜑𝐴 ∈ 𝒫 𝑋)
133, 12ffvelrnd 6834 1 (𝜑 → (𝑂𝐴) ∈ (0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2114  Vcvv 3469  wss 3908  𝒫 cpw 4511   cuni 4813  dom cdm 5532  cfv 6334  (class class class)co 7140  0cc0 10526  +∞cpnf 10661  [,]cicc 12729  OutMeascome 43067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-ome 43068
This theorem is referenced by:  caragen0  43084  omexrcl  43085  caragenunidm  43086  omessre  43088  caragenuncllem  43090  caragendifcl  43092  omeunle  43094  omeiunle  43095  omeiunltfirp  43097  carageniuncllem2  43100  carageniuncl  43101  caratheodorylem1  43104  caratheodorylem2  43105  omege0  43111
  Copyright terms: Public domain W3C validator