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Theorem elcarsg 33958
Description: Property of being a Caratheodory measurable set. (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
elcarsg (πœ‘ β†’ (𝐴 ∈ (toCaraSigaβ€˜π‘€) ↔ (𝐴 βŠ† 𝑂 ∧ βˆ€π‘’ ∈ 𝒫 𝑂((π‘€β€˜(𝑒 ∩ 𝐴)) +𝑒 (π‘€β€˜(𝑒 βˆ– 𝐴))) = (π‘€β€˜π‘’))))
Distinct variable groups:   𝑒,𝑀   𝑒,𝑂   πœ‘,𝑒   𝐴,𝑒
Allowed substitution hint:   𝑉(𝑒)

Proof of Theorem elcarsg
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 carsgval.1 . . . 4 (πœ‘ β†’ 𝑂 ∈ 𝑉)
2 carsgval.2 . . . 4 (πœ‘ β†’ 𝑀:𝒫 π‘‚βŸΆ(0[,]+∞))
31, 2carsgval 33956 . . 3 (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‘’)})
43eleq2d 2815 . 2 (πœ‘ β†’ (𝐴 ∈ (toCaraSigaβ€˜π‘€) ↔ 𝐴 ∈ {π‘Ž ∈ 𝒫 𝑂 ∣ βˆ€π‘’ ∈ 𝒫 𝑂((π‘€β€˜(𝑒 ∩ π‘Ž)) +𝑒 (π‘€β€˜(𝑒 βˆ– π‘Ž))) = (π‘€β€˜π‘’)}))
5 ineq2 4208 . . . . . . . 8 (π‘Ž = 𝐴 β†’ (𝑒 ∩ π‘Ž) = (𝑒 ∩ 𝐴))
65fveq2d 6906 . . . . . . 7 (π‘Ž = 𝐴 β†’ (π‘€β€˜(𝑒 ∩ π‘Ž)) = (π‘€β€˜(𝑒 ∩ 𝐴)))
7 difeq2 4116 . . . . . . . 8 (π‘Ž = 𝐴 β†’ (𝑒 βˆ– π‘Ž) = (𝑒 βˆ– 𝐴))
87fveq2d 6906 . . . . . . 7 (π‘Ž = 𝐴 β†’ (π‘€β€˜(𝑒 βˆ– π‘Ž)) = (π‘€β€˜(𝑒 βˆ– 𝐴)))
96, 8oveq12d 7444 . . . . . 6 (π‘Ž = 𝐴 β†’ ((π‘€β€˜(𝑒 ∩ π‘Ž)) +𝑒 (π‘€β€˜(𝑒 βˆ– π‘Ž))) = ((π‘€β€˜(𝑒 ∩ 𝐴)) +𝑒 (π‘€β€˜(𝑒 βˆ– 𝐴))))
109eqeq1d 2730 . . . . 5 (π‘Ž = 𝐴 β†’ (((π‘€β€˜(𝑒 ∩ π‘Ž)) +𝑒 (π‘€β€˜(𝑒 βˆ– π‘Ž))) = (π‘€β€˜π‘’) ↔ ((π‘€β€˜(𝑒 ∩ 𝐴)) +𝑒 (π‘€β€˜(𝑒 βˆ– 𝐴))) = (π‘€β€˜π‘’)))
1110ralbidv 3175 . . . 4 (π‘Ž = 𝐴 β†’ (βˆ€π‘’ ∈ 𝒫 𝑂((π‘€β€˜(𝑒 ∩ π‘Ž)) +𝑒 (π‘€β€˜(𝑒 βˆ– π‘Ž))) = (π‘€β€˜π‘’) ↔ βˆ€π‘’ ∈ 𝒫 𝑂((π‘€β€˜(𝑒 ∩ 𝐴)) +𝑒 (π‘€β€˜(𝑒 βˆ– 𝐴))) = (π‘€β€˜π‘’)))
1211elrab 3684 . . 3 (𝐴 ∈ {π‘Ž ∈ 𝒫 𝑂 ∣ βˆ€π‘’ ∈ 𝒫 𝑂((π‘€β€˜(𝑒 ∩ π‘Ž)) +𝑒 (π‘€β€˜(𝑒 βˆ– π‘Ž))) = (π‘€β€˜π‘’)} ↔ (𝐴 ∈ 𝒫 𝑂 ∧ βˆ€π‘’ ∈ 𝒫 𝑂((π‘€β€˜(𝑒 ∩ 𝐴)) +𝑒 (π‘€β€˜(𝑒 βˆ– 𝐴))) = (π‘€β€˜π‘’)))
13 elex 3492 . . . . . 6 (𝐴 ∈ 𝒫 𝑂 β†’ 𝐴 ∈ V)
1413a1i 11 . . . . 5 (πœ‘ β†’ (𝐴 ∈ 𝒫 𝑂 β†’ 𝐴 ∈ V))
151adantr 479 . . . . . . 7 ((πœ‘ ∧ 𝐴 βŠ† 𝑂) β†’ 𝑂 ∈ 𝑉)
16 simpr 483 . . . . . . 7 ((πœ‘ ∧ 𝐴 βŠ† 𝑂) β†’ 𝐴 βŠ† 𝑂)
1715, 16ssexd 5328 . . . . . 6 ((πœ‘ ∧ 𝐴 βŠ† 𝑂) β†’ 𝐴 ∈ V)
1817ex 411 . . . . 5 (πœ‘ β†’ (𝐴 βŠ† 𝑂 β†’ 𝐴 ∈ V))
19 elpwg 4609 . . . . . 6 (𝐴 ∈ V β†’ (𝐴 ∈ 𝒫 𝑂 ↔ 𝐴 βŠ† 𝑂))
2019a1i 11 . . . . 5 (πœ‘ β†’ (𝐴 ∈ V β†’ (𝐴 ∈ 𝒫 𝑂 ↔ 𝐴 βŠ† 𝑂)))
2114, 18, 20pm5.21ndd 378 . . . 4 (πœ‘ β†’ (𝐴 ∈ 𝒫 𝑂 ↔ 𝐴 βŠ† 𝑂))
2221anbi1d 629 . . 3 (πœ‘ β†’ ((𝐴 ∈ 𝒫 𝑂 ∧ βˆ€π‘’ ∈ 𝒫 𝑂((π‘€β€˜(𝑒 ∩ 𝐴)) +𝑒 (π‘€β€˜(𝑒 βˆ– 𝐴))) = (π‘€β€˜π‘’)) ↔ (𝐴 βŠ† 𝑂 ∧ βˆ€π‘’ ∈ 𝒫 𝑂((π‘€β€˜(𝑒 ∩ 𝐴)) +𝑒 (π‘€β€˜(𝑒 βˆ– 𝐴))) = (π‘€β€˜π‘’))))
2312, 22bitrid 282 . 2 (πœ‘ β†’ (𝐴 ∈ {π‘Ž ∈ 𝒫 𝑂 ∣ βˆ€π‘’ ∈ 𝒫 𝑂((π‘€β€˜(𝑒 ∩ π‘Ž)) +𝑒 (π‘€β€˜(𝑒 βˆ– π‘Ž))) = (π‘€β€˜π‘’)} ↔ (𝐴 βŠ† 𝑂 ∧ βˆ€π‘’ ∈ 𝒫 𝑂((π‘€β€˜(𝑒 ∩ 𝐴)) +𝑒 (π‘€β€˜(𝑒 βˆ– 𝐴))) = (π‘€β€˜π‘’))))
244, 23bitrd 278 1 (πœ‘ β†’ (𝐴 ∈ (toCaraSigaβ€˜π‘€) ↔ (𝐴 βŠ† 𝑂 ∧ βˆ€π‘’ ∈ 𝒫 𝑂((π‘€β€˜(𝑒 ∩ 𝐴)) +𝑒 (π‘€β€˜(𝑒 βˆ– 𝐴))) = (π‘€β€˜π‘’))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  {crab 3430  Vcvv 3473   βˆ– cdif 3946   ∩ cin 3948   βŠ† wss 3949  π’« cpw 4606  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426  0cc0 11146  +∞cpnf 11283   +𝑒 cxad 13130  [,]cicc 13367  toCaraSigaccarsg 33954
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
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 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-carsg 33955
This theorem is referenced by:  baselcarsg  33959  0elcarsg  33960  difelcarsg  33963  inelcarsg  33964  carsgclctunlem1  33970  carsgclctunlem2  33972  carsgclctun  33974
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