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Theorem caragenss 40022
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 3666 . . 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 40011 . . . 4 (𝑂 ∈ OutMeas → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
64, 5eqtrd 2655 . . 3 (𝑂 ∈ OutMeas → 𝑆 = {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
7 omedm 40017 . . 3 (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 dom 𝑂)
86, 7sseq12d 3613 . 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 247 1 (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wral 2907  {crab 2911  cdif 3552  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402  dom cdm 5074  cfv 5847  (class class class)co 6604   +𝑒 cxad 11888  OutMeascome 40007  CaraGenccaragen 40009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-ome 40008  df-caragen 40010
This theorem is referenced by:  caragensspw  40027  caragenuni  40029  caragendifcl  40032  caratheodorylem1  40044  caratheodorylem2  40045  dmvon  40124  voncmpl  40139  vonmblss  40158
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