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Theorem carsgcl 33833
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 33832 . 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 4072 . 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 4031 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 3055  {crab 3426   βˆ– cdif 3940   ∩ cin 3942   βŠ† wss 3943  π’« cpw 4597  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404  0cc0 11109  +∞cpnf 11246   +𝑒 cxad 13093  [,]cicc 13330  toCaraSigaccarsg 33830
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-carsg 33831
This theorem is referenced by:  carsguni  33837  elcarsgss  33838  carsggect  33847  carsgsiga  33851  omsmeas  33852
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