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Theorem caragenss 43717
Description: The sigma-algebra generated from an outer measure, by the Caratheodory's construction, is a subset of the domain of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
caragenss.1 𝑆 = (CaraGen‘𝑂)
Assertion
Ref Expression
caragenss (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)

Proof of Theorem caragenss
Dummy variables 𝑎 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3993 . . 3 {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)} ⊆ 𝒫 dom 𝑂
21a1i 11 . 2 (𝑂 ∈ OutMeas → {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)} ⊆ 𝒫 dom 𝑂)
3 caragenss.1 . . . . 5 𝑆 = (CaraGen‘𝑂)
43a1i 11 . . . 4 (𝑂 ∈ OutMeas → 𝑆 = (CaraGen‘𝑂))
5 caragenval 43706 . . . 4 (𝑂 ∈ OutMeas → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
64, 5eqtrd 2777 . . 3 (𝑂 ∈ OutMeas → 𝑆 = {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
7 omedm 43712 . . 3 (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 dom 𝑂)
86, 7sseq12d 3934 . 2 (𝑂 ∈ OutMeas → (𝑆 ⊆ dom 𝑂 ↔ {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)} ⊆ 𝒫 dom 𝑂))
92, 8mpbird 260 1 (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  wral 3061  {crab 3065  cdif 3863  cin 3865  wss 3866  𝒫 cpw 4513   cuni 4819  dom cdm 5551  cfv 6380  (class class class)co 7213   +𝑒 cxad 12702  OutMeascome 43702  CaraGenccaragen 43704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-ov 7216  df-ome 43703  df-caragen 43705
This theorem is referenced by:  caragensspw  43722  caragenuni  43724  caragendifcl  43727  caratheodorylem1  43739  caratheodorylem2  43740  dmvon  43819  voncmpl  43834  vonmblss  43853
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