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Theorem carsgcl 31589
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 31588 . 2 (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)})
4 ssrab2 4042 . 2 {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)} ⊆ 𝒫 𝑂
53, 4eqsstrdi 4007 1 (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  wral 3133  {crab 3137  cdif 3916  cin 3918  wss 3919  𝒫 cpw 4522  wf 6340  cfv 6344  (class class class)co 7146  0cc0 10531  +∞cpnf 10666   +𝑒 cxad 12500  [,]cicc 12736  toCaraSigaccarsg 31586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-carsg 31587
This theorem is referenced by:  carsguni  31593  elcarsgss  31594  carsggect  31603  carsgsiga  31607  omsmeas  31608
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