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Theorem caragenss 41242
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 3828 . . 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 41231 . . . 4 (𝑂 ∈ OutMeas → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
64, 5eqtrd 2794 . . 3 (𝑂 ∈ OutMeas → 𝑆 = {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
7 omedm 41237 . . 3 (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 dom 𝑂)
86, 7sseq12d 3775 . 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 1632  wcel 2139  wral 3050  {crab 3054  cdif 3712  cin 3714  wss 3715  𝒫 cpw 4302   cuni 4588  dom cdm 5266  cfv 6049  (class class class)co 6814   +𝑒 cxad 12157  OutMeascome 41227  CaraGenccaragen 41229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-ome 41228  df-caragen 41230
This theorem is referenced by:  caragensspw  41247  caragenuni  41249  caragendifcl  41252  caratheodorylem1  41264  caratheodorylem2  41265  dmvon  41344  voncmpl  41359  vonmblss  41378
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