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Theorem caragensspw 42668
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 42663 . . . 4 (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
41, 3syl 17 . . 3 (𝜑𝑆 ⊆ dom 𝑂)
5 pwuni 4866 . . . 4 dom 𝑂 ⊆ 𝒫 dom 𝑂
65a1i 11 . . 3 (𝜑 → dom 𝑂 ⊆ 𝒫 dom 𝑂)
74, 6sstrd 3974 . 2 (𝜑𝑆 ⊆ 𝒫 dom 𝑂)
8 caragensspw.x . . . . 5 𝑋 = dom 𝑂
98pweqi 4539 . . . 4 𝒫 𝑋 = 𝒫 dom 𝑂
109eqcomi 2827 . . 3 𝒫 dom 𝑂 = 𝒫 𝑋
1110a1i 11 . 2 (𝜑 → 𝒫 dom 𝑂 = 𝒫 𝑋)
127, 11sseqtrd 4004 1 (𝜑𝑆 ⊆ 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wss 3933  𝒫 cpw 4535   cuni 4830  dom cdm 5548  cfv 6348  OutMeascome 42648  CaraGenccaragen 42650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-ome 42649  df-caragen 42651
This theorem is referenced by:  caratheodorylem2  42686
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