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Theorem caragensspw 41510
 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 41505 . . . 4 (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
41, 3syl 17 . . 3 (𝜑𝑆 ⊆ dom 𝑂)
5 pwuni 4696 . . . 4 dom 𝑂 ⊆ 𝒫 dom 𝑂
65a1i 11 . . 3 (𝜑 → dom 𝑂 ⊆ 𝒫 dom 𝑂)
74, 6sstrd 3837 . 2 (𝜑𝑆 ⊆ 𝒫 dom 𝑂)
8 caragensspw.x . . . . 5 𝑋 = dom 𝑂
98pweqi 4382 . . . 4 𝒫 𝑋 = 𝒫 dom 𝑂
109eqcomi 2834 . . 3 𝒫 dom 𝑂 = 𝒫 𝑋
1110a1i 11 . 2 (𝜑 → 𝒫 dom 𝑂 = 𝒫 𝑋)
127, 11sseqtrd 3866 1 (𝜑𝑆 ⊆ 𝒫 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1656   ∈ wcel 2164   ⊆ wss 3798  𝒫 cpw 4378  ∪ cuni 4658  dom cdm 5342  ‘cfv 6123  OutMeascome 41490  CaraGenccaragen 41492 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ov 6908  df-ome 41491  df-caragen 41493 This theorem is referenced by:  caratheodorylem2  41528
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