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Theorem elcarsg 33834
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 33832 . . 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 2813 . 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 4201 . . . . . . . 8 (π‘Ž = 𝐴 β†’ (𝑒 ∩ π‘Ž) = (𝑒 ∩ 𝐴))
65fveq2d 6889 . . . . . . 7 (π‘Ž = 𝐴 β†’ (π‘€β€˜(𝑒 ∩ π‘Ž)) = (π‘€β€˜(𝑒 ∩ 𝐴)))
7 difeq2 4111 . . . . . . . 8 (π‘Ž = 𝐴 β†’ (𝑒 βˆ– π‘Ž) = (𝑒 βˆ– 𝐴))
87fveq2d 6889 . . . . . . 7 (π‘Ž = 𝐴 β†’ (π‘€β€˜(𝑒 βˆ– π‘Ž)) = (π‘€β€˜(𝑒 βˆ– 𝐴)))
96, 8oveq12d 7423 . . . . . 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 2728 . . . . 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 3171 . . . 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 3678 . . 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 3487 . . . . . 6 (𝐴 ∈ 𝒫 𝑂 β†’ 𝐴 ∈ V)
1413a1i 11 . . . . 5 (πœ‘ β†’ (𝐴 ∈ 𝒫 𝑂 β†’ 𝐴 ∈ V))
151adantr 480 . . . . . . 7 ((πœ‘ ∧ 𝐴 βŠ† 𝑂) β†’ 𝑂 ∈ 𝑉)
16 simpr 484 . . . . . . 7 ((πœ‘ ∧ 𝐴 βŠ† 𝑂) β†’ 𝐴 βŠ† 𝑂)
1715, 16ssexd 5317 . . . . . 6 ((πœ‘ ∧ 𝐴 βŠ† 𝑂) β†’ 𝐴 ∈ V)
1817ex 412 . . . . 5 (πœ‘ β†’ (𝐴 βŠ† 𝑂 β†’ 𝐴 ∈ V))
19 elpwg 4600 . . . . . 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 379 . . . 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 283 . 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 279 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 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  {crab 3426  Vcvv 3468   βˆ– cdif 3940   ∩ cin 3942   βŠ† wss 3943  π’« cpw 4597  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405  0cc0 11112  +∞cpnf 11249   +𝑒 cxad 13096  [,]cicc 13333  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 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-carsg 33831
This theorem is referenced by:  baselcarsg  33835  0elcarsg  33836  difelcarsg  33839  inelcarsg  33840  carsgclctunlem1  33846  carsgclctunlem2  33848  carsgclctun  33850
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