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Theorem caragenss 42663
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 4053 . . 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 42652 . . . 4 (𝑂 ∈ OutMeas → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
64, 5eqtrd 2853 . . 3 (𝑂 ∈ OutMeas → 𝑆 = {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
7 omedm 42658 . . 3 (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 dom 𝑂)
86, 7sseq12d 3997 . 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 258 1 (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wral 3135  {crab 3139  cdif 3930  cin 3932  wss 3933  𝒫 cpw 4535   cuni 4830  dom cdm 5548  cfv 6348  (class class class)co 7145   +𝑒 cxad 12493  OutMeascome 42648  CaraGenccaragen 42650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-ome 42649  df-caragen 42651
This theorem is referenced by:  caragensspw  42668  caragenuni  42670  caragendifcl  42673  caratheodorylem1  42685  caratheodorylem2  42686  dmvon  42765  voncmpl  42780  vonmblss  42799
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