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Theorem elcarsg 32945
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 32943 . . 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 2824 . 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 4171 . . . . . . . 8 (π‘Ž = 𝐴 β†’ (𝑒 ∩ π‘Ž) = (𝑒 ∩ 𝐴))
65fveq2d 6851 . . . . . . 7 (π‘Ž = 𝐴 β†’ (π‘€β€˜(𝑒 ∩ π‘Ž)) = (π‘€β€˜(𝑒 ∩ 𝐴)))
7 difeq2 4081 . . . . . . . 8 (π‘Ž = 𝐴 β†’ (𝑒 βˆ– π‘Ž) = (𝑒 βˆ– 𝐴))
87fveq2d 6851 . . . . . . 7 (π‘Ž = 𝐴 β†’ (π‘€β€˜(𝑒 βˆ– π‘Ž)) = (π‘€β€˜(𝑒 βˆ– 𝐴)))
96, 8oveq12d 7380 . . . . . 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 2739 . . . . 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 3650 . . 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 3466 . . . . . 6 (𝐴 ∈ 𝒫 𝑂 β†’ 𝐴 ∈ V)
1413a1i 11 . . . . 5 (πœ‘ β†’ (𝐴 ∈ 𝒫 𝑂 β†’ 𝐴 ∈ V))
151adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝐴 βŠ† 𝑂) β†’ 𝑂 ∈ 𝑉)
16 simpr 486 . . . . . . 7 ((πœ‘ ∧ 𝐴 βŠ† 𝑂) β†’ 𝐴 βŠ† 𝑂)
1715, 16ssexd 5286 . . . . . 6 ((πœ‘ ∧ 𝐴 βŠ† 𝑂) β†’ 𝐴 ∈ V)
1817ex 414 . . . . 5 (πœ‘ β†’ (𝐴 βŠ† 𝑂 β†’ 𝐴 ∈ V))
19 elpwg 4568 . . . . . 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 381 . . . 4 (πœ‘ β†’ (𝐴 ∈ 𝒫 𝑂 ↔ 𝐴 βŠ† 𝑂))
2221anbi1d 631 . . 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3410  Vcvv 3448   βˆ– cdif 3912   ∩ cin 3914   βŠ† wss 3915  π’« cpw 4565  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362  0cc0 11058  +∞cpnf 11193   +𝑒 cxad 13038  [,]cicc 13274  toCaraSigaccarsg 32941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-carsg 32942
This theorem is referenced by:  baselcarsg  32946  0elcarsg  32947  difelcarsg  32950  inelcarsg  32951  carsgclctunlem1  32957  carsgclctunlem2  32959  carsgclctun  32961
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