Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  caragensspw Structured version   Visualization version   GIF version

Theorem caragensspw 44824
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 44819 . . . 4 (𝑂 ∈ OutMeas β†’ 𝑆 βŠ† dom 𝑂)
41, 3syl 17 . . 3 (πœ‘ β†’ 𝑆 βŠ† dom 𝑂)
5 pwuni 4911 . . . 4 dom 𝑂 βŠ† 𝒫 βˆͺ dom 𝑂
65a1i 11 . . 3 (πœ‘ β†’ dom 𝑂 βŠ† 𝒫 βˆͺ dom 𝑂)
74, 6sstrd 3959 . 2 (πœ‘ β†’ 𝑆 βŠ† 𝒫 βˆͺ dom 𝑂)
8 caragensspw.x . . . . 5 𝑋 = βˆͺ dom 𝑂
98pweqi 4581 . . . 4 𝒫 𝑋 = 𝒫 βˆͺ dom 𝑂
109eqcomi 2746 . . 3 𝒫 βˆͺ dom 𝑂 = 𝒫 𝑋
1110a1i 11 . 2 (πœ‘ β†’ 𝒫 βˆͺ dom 𝑂 = 𝒫 𝑋)
127, 11sseqtrd 3989 1 (πœ‘ β†’ 𝑆 βŠ† 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107   βŠ† wss 3915  π’« cpw 4565  βˆͺ cuni 4870  dom cdm 5638  β€˜cfv 6501  OutMeascome 44804  CaraGenccaragen 44806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ov 7365  df-ome 44805  df-caragen 44807
This theorem is referenced by:  caratheodorylem2  44842
  Copyright terms: Public domain W3C validator