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Theorem elcarsg 31637
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 31635 . . 3 (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)})
43eleq2d 2899 . 2 (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ 𝐴 ∈ {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)}))
5 ineq2 4157 . . . . . . . 8 (𝑎 = 𝐴 → (𝑒𝑎) = (𝑒𝐴))
65fveq2d 6656 . . . . . . 7 (𝑎 = 𝐴 → (𝑀‘(𝑒𝑎)) = (𝑀‘(𝑒𝐴)))
7 difeq2 4068 . . . . . . . 8 (𝑎 = 𝐴 → (𝑒𝑎) = (𝑒𝐴))
87fveq2d 6656 . . . . . . 7 (𝑎 = 𝐴 → (𝑀‘(𝑒𝑎)) = (𝑀‘(𝑒𝐴)))
96, 8oveq12d 7158 . . . . . 6 (𝑎 = 𝐴 → ((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = ((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))))
109eqeq1d 2824 . . . . 5 (𝑎 = 𝐴 → (((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒) ↔ ((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒)))
1110ralbidv 3187 . . . 4 (𝑎 = 𝐴 → (∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒) ↔ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒)))
1211elrab 3655 . . 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 3487 . . . . . 6 (𝐴 ∈ 𝒫 𝑂𝐴 ∈ V)
1413a1i 11 . . . . 5 (𝜑 → (𝐴 ∈ 𝒫 𝑂𝐴 ∈ V))
151adantr 484 . . . . . . 7 ((𝜑𝐴𝑂) → 𝑂𝑉)
16 simpr 488 . . . . . . 7 ((𝜑𝐴𝑂) → 𝐴𝑂)
1715, 16ssexd 5204 . . . . . 6 ((𝜑𝐴𝑂) → 𝐴 ∈ V)
1817ex 416 . . . . 5 (𝜑 → (𝐴𝑂𝐴 ∈ V))
19 elpwg 4514 . . . . . 6 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑂𝐴𝑂))
2019a1i 11 . . . . 5 (𝜑 → (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑂𝐴𝑂)))
2114, 18, 20pm5.21ndd 384 . . . 4 (𝜑 → (𝐴 ∈ 𝒫 𝑂𝐴𝑂))
2221anbi1d 632 . . 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 286 . 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 282 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 209  wa 399   = wceq 1538  wcel 2114  wral 3130  {crab 3134  Vcvv 3469  cdif 3905  cin 3907  wss 3908  𝒫 cpw 4511  wf 6330  cfv 6334  (class class class)co 7140  0cc0 10526  +∞cpnf 10661   +𝑒 cxad 12493  [,]cicc 12729  toCaraSigaccarsg 31633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-carsg 31634
This theorem is referenced by:  baselcarsg  31638  0elcarsg  31639  difelcarsg  31642  inelcarsg  31643  carsgclctunlem1  31649  carsgclctunlem2  31651  carsgclctun  31653
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