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Theorem carsgcl 33929
Description: Closure of the Caratheodory measurable sets. (Contributed by Thierry Arnoux, 17-May-2020.)
Hypotheses
Ref Expression
carsgval.1 (πœ‘ β†’ 𝑂 ∈ 𝑉)
carsgval.2 (πœ‘ β†’ 𝑀:𝒫 π‘‚βŸΆ(0[,]+∞))
Assertion
Ref Expression
carsgcl (πœ‘ β†’ (toCaraSigaβ€˜π‘€) βŠ† 𝒫 𝑂)

Proof of Theorem carsgcl
Dummy variables π‘Ž 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 carsgval.1 . . 3 (πœ‘ β†’ 𝑂 ∈ 𝑉)
2 carsgval.2 . . 3 (πœ‘ β†’ 𝑀:𝒫 π‘‚βŸΆ(0[,]+∞))
31, 2carsgval 33928 . 2 (πœ‘ β†’ (toCaraSigaβ€˜π‘€) = {π‘Ž ∈ 𝒫 𝑂 ∣ βˆ€π‘’ ∈ 𝒫 𝑂((π‘€β€˜(𝑒 ∩ π‘Ž)) +𝑒 (π‘€β€˜(𝑒 βˆ– π‘Ž))) = (π‘€β€˜π‘’)})
4 ssrab2 4075 . 2 {π‘Ž ∈ 𝒫 𝑂 ∣ βˆ€π‘’ ∈ 𝒫 𝑂((π‘€β€˜(𝑒 ∩ π‘Ž)) +𝑒 (π‘€β€˜(𝑒 βˆ– π‘Ž))) = (π‘€β€˜π‘’)} βŠ† 𝒫 𝑂
53, 4eqsstrdi 4034 1 (πœ‘ β†’ (toCaraSigaβ€˜π‘€) βŠ† 𝒫 𝑂)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  βˆ€wral 3057  {crab 3428   βˆ– cdif 3944   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4604  βŸΆwf 6547  β€˜cfv 6551  (class class class)co 7424  0cc0 11144  +∞cpnf 11281   +𝑒 cxad 13128  [,]cicc 13365  toCaraSigaccarsg 33926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-carsg 33927
This theorem is referenced by:  carsguni  33933  elcarsgss  33934  carsggect  33943  carsgsiga  33947  omsmeas  33948
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