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Theorem caragensspw 46524
Description: The sigma-algebra generated from an outer measure, by the Caratheodory's construction, is a subset of the power set of the base set of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
caragensspw.o (𝜑𝑂 ∈ OutMeas)
caragensspw.x 𝑋 = dom 𝑂
caragensspw.s 𝑆 = (CaraGen‘𝑂)
Assertion
Ref Expression
caragensspw (𝜑𝑆 ⊆ 𝒫 𝑋)

Proof of Theorem caragensspw
StepHypRef Expression
1 caragensspw.o . . . 4 (𝜑𝑂 ∈ OutMeas)
2 caragensspw.s . . . . 5 𝑆 = (CaraGen‘𝑂)
32caragenss 46519 . . . 4 (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
41, 3syl 17 . . 3 (𝜑𝑆 ⊆ dom 𝑂)
5 pwuni 4945 . . . 4 dom 𝑂 ⊆ 𝒫 dom 𝑂
65a1i 11 . . 3 (𝜑 → dom 𝑂 ⊆ 𝒫 dom 𝑂)
74, 6sstrd 3994 . 2 (𝜑𝑆 ⊆ 𝒫 dom 𝑂)
8 caragensspw.x . . . . 5 𝑋 = dom 𝑂
98pweqi 4616 . . . 4 𝒫 𝑋 = 𝒫 dom 𝑂
109eqcomi 2746 . . 3 𝒫 dom 𝑂 = 𝒫 𝑋
1110a1i 11 . 2 (𝜑 → 𝒫 dom 𝑂 = 𝒫 𝑋)
127, 11sseqtrd 4020 1 (𝜑𝑆 ⊆ 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wss 3951  𝒫 cpw 4600   cuni 4907  dom cdm 5685  cfv 6561  OutMeascome 46504  CaraGenccaragen 46506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-ome 46505  df-caragen 46507
This theorem is referenced by:  caratheodorylem2  46542
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