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Theorem elcarsg 31462
Description: Property of being a Caratheodory measurable set. (Contributed by Thierry Arnoux, 17-May-2020.)
Hypotheses
Ref Expression
carsgval.1 (𝜑𝑂𝑉)
carsgval.2 (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))
Assertion
Ref Expression
elcarsg (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ (𝐴𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒))))
Distinct variable groups:   𝑒,𝑀   𝑒,𝑂   𝜑,𝑒   𝐴,𝑒
Allowed substitution hint:   𝑉(𝑒)

Proof of Theorem elcarsg
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 carsgval.1 . . . 4 (𝜑𝑂𝑉)
2 carsgval.2 . . . 4 (𝜑𝑀:𝒫 𝑂⟶(0[,]+∞))
31, 2carsgval 31460 . . 3 (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)})
43eleq2d 2895 . 2 (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ 𝐴 ∈ {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)}))
5 ineq2 4180 . . . . . . . 8 (𝑎 = 𝐴 → (𝑒𝑎) = (𝑒𝐴))
65fveq2d 6667 . . . . . . 7 (𝑎 = 𝐴 → (𝑀‘(𝑒𝑎)) = (𝑀‘(𝑒𝐴)))
7 difeq2 4090 . . . . . . . 8 (𝑎 = 𝐴 → (𝑒𝑎) = (𝑒𝐴))
87fveq2d 6667 . . . . . . 7 (𝑎 = 𝐴 → (𝑀‘(𝑒𝑎)) = (𝑀‘(𝑒𝐴)))
96, 8oveq12d 7163 . . . . . 6 (𝑎 = 𝐴 → ((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = ((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))))
109eqeq1d 2820 . . . . 5 (𝑎 = 𝐴 → (((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒) ↔ ((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒)))
1110ralbidv 3194 . . . 4 (𝑎 = 𝐴 → (∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒) ↔ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒)))
1211elrab 3677 . . 3 (𝐴 ∈ {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)} ↔ (𝐴 ∈ 𝒫 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒)))
13 elex 3510 . . . . . 6 (𝐴 ∈ 𝒫 𝑂𝐴 ∈ V)
1413a1i 11 . . . . 5 (𝜑 → (𝐴 ∈ 𝒫 𝑂𝐴 ∈ V))
151adantr 481 . . . . . . 7 ((𝜑𝐴𝑂) → 𝑂𝑉)
16 simpr 485 . . . . . . 7 ((𝜑𝐴𝑂) → 𝐴𝑂)
1715, 16ssexd 5219 . . . . . 6 ((𝜑𝐴𝑂) → 𝐴 ∈ V)
1817ex 413 . . . . 5 (𝜑 → (𝐴𝑂𝐴 ∈ V))
19 elpwg 4541 . . . . . 6 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑂𝐴𝑂))
2019a1i 11 . . . . 5 (𝜑 → (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑂𝐴𝑂)))
2114, 18, 20pm5.21ndd 381 . . . 4 (𝜑 → (𝐴 ∈ 𝒫 𝑂𝐴𝑂))
2221anbi1d 629 . . 3 (𝜑 → ((𝐴 ∈ 𝒫 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒)) ↔ (𝐴𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒))))
2312, 22syl5bb 284 . 2 (𝜑 → (𝐴 ∈ {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)} ↔ (𝐴𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒))))
244, 23bitrd 280 1 (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ (𝐴𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492  cdif 3930  cin 3932  wss 3933  𝒫 cpw 4535  wf 6344  cfv 6348  (class class class)co 7145  0cc0 10525  +∞cpnf 10660   +𝑒 cxad 12493  [,]cicc 12729  toCaraSigaccarsg 31458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-carsg 31459
This theorem is referenced by:  baselcarsg  31463  0elcarsg  31464  difelcarsg  31467  inelcarsg  31468  carsgclctunlem1  31474  carsgclctunlem2  31476  carsgclctun  31478
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