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Theorem caragensspw 42935
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 42930 . . . 4 (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
41, 3syl 17 . . 3 (𝜑𝑆 ⊆ dom 𝑂)
5 pwuni 4847 . . . 4 dom 𝑂 ⊆ 𝒫 dom 𝑂
65a1i 11 . . 3 (𝜑 → dom 𝑂 ⊆ 𝒫 dom 𝑂)
74, 6sstrd 3952 . 2 (𝜑𝑆 ⊆ 𝒫 dom 𝑂)
8 caragensspw.x . . . . 5 𝑋 = dom 𝑂
98pweqi 4529 . . . 4 𝒫 𝑋 = 𝒫 dom 𝑂
109eqcomi 2829 . . 3 𝒫 dom 𝑂 = 𝒫 𝑋
1110a1i 11 . 2 (𝜑 → 𝒫 dom 𝑂 = 𝒫 𝑋)
127, 11sseqtrd 3982 1 (𝜑𝑆 ⊆ 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  wss 3909  𝒫 cpw 4511   cuni 4810  dom cdm 5527  cfv 6327  OutMeascome 42915  CaraGenccaragen 42917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-fv 6335  df-ov 7132  df-ome 42916  df-caragen 42918
This theorem is referenced by:  caratheodorylem2  42953
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