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Theorem carsgcl 34286
Description: Closure of the Caratheodory measurable sets. (Contributed by Thierry Arnoux, 17-May-2020.)
Hypotheses
Ref Expression
carsgval.1 (𝜑𝑂𝑉)
carsgval.2 (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))
Assertion
Ref Expression
carsgcl (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)

Proof of Theorem carsgcl
Dummy variables 𝑎 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 carsgval.1 . . 3 (𝜑𝑂𝑉)
2 carsgval.2 . . 3 (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))
31, 2carsgval 34285 . 2 (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)})
4 ssrab2 4090 . 2 {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)} ⊆ 𝒫 𝑂
53, 4eqsstrdi 4050 1 (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wral 3059  {crab 3433  cdif 3960  cin 3962  wss 3963  𝒫 cpw 4605  wf 6559  cfv 6563  (class class class)co 7431  0cc0 11153  +∞cpnf 11290   +𝑒 cxad 13150  [,]cicc 13387  toCaraSigaccarsg 34283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-carsg 34284
This theorem is referenced by:  carsguni  34290  elcarsgss  34291  carsggect  34300  carsgsiga  34304  omsmeas  34305
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