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Theorem caragensspw 45959
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 (πœ‘ β†’ 𝑂 ∈ OutMeas)
caragensspw.x 𝑋 = βˆͺ dom 𝑂
caragensspw.s 𝑆 = (CaraGenβ€˜π‘‚)
Assertion
Ref Expression
caragensspw (πœ‘ β†’ 𝑆 βŠ† 𝒫 𝑋)

Proof of Theorem caragensspw
StepHypRef Expression
1 caragensspw.o . . . 4 (πœ‘ β†’ 𝑂 ∈ OutMeas)
2 caragensspw.s . . . . 5 𝑆 = (CaraGenβ€˜π‘‚)
32caragenss 45954 . . . 4 (𝑂 ∈ OutMeas β†’ 𝑆 βŠ† dom 𝑂)
41, 3syl 17 . . 3 (πœ‘ β†’ 𝑆 βŠ† dom 𝑂)
5 pwuni 4943 . . . 4 dom 𝑂 βŠ† 𝒫 βˆͺ dom 𝑂
65a1i 11 . . 3 (πœ‘ β†’ dom 𝑂 βŠ† 𝒫 βˆͺ dom 𝑂)
74, 6sstrd 3983 . 2 (πœ‘ β†’ 𝑆 βŠ† 𝒫 βˆͺ dom 𝑂)
8 caragensspw.x . . . . 5 𝑋 = βˆͺ dom 𝑂
98pweqi 4614 . . . 4 𝒫 𝑋 = 𝒫 βˆͺ dom 𝑂
109eqcomi 2734 . . 3 𝒫 βˆͺ dom 𝑂 = 𝒫 𝑋
1110a1i 11 . 2 (πœ‘ β†’ 𝒫 βˆͺ dom 𝑂 = 𝒫 𝑋)
127, 11sseqtrd 4013 1 (πœ‘ β†’ 𝑆 βŠ† 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098   βŠ† wss 3940  π’« cpw 4598  βˆͺ cuni 4903  dom cdm 5672  β€˜cfv 6542  OutMeascome 45939  CaraGenccaragen 45941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7418  df-ome 45940  df-caragen 45942
This theorem is referenced by:  caratheodorylem2  45977
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