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Theorem elcarsg 30707
Description: Property of being a Catatheodory 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 30705 . . 3 (𝜑 → (toCaraSiga‘𝑀) = {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)})
43eleq2d 2836 . 2 (𝜑 → (𝐴 ∈ (toCaraSiga‘𝑀) ↔ 𝐴 ∈ {𝑎 ∈ 𝒫 𝑂 ∣ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒)}))
5 ineq2 3959 . . . . . . . 8 (𝑎 = 𝐴 → (𝑒𝑎) = (𝑒𝐴))
65fveq2d 6336 . . . . . . 7 (𝑎 = 𝐴 → (𝑀‘(𝑒𝑎)) = (𝑀‘(𝑒𝐴)))
7 difeq2 3873 . . . . . . . 8 (𝑎 = 𝐴 → (𝑒𝑎) = (𝑒𝐴))
87fveq2d 6336 . . . . . . 7 (𝑎 = 𝐴 → (𝑀‘(𝑒𝑎)) = (𝑀‘(𝑒𝐴)))
96, 8oveq12d 6811 . . . . . 6 (𝑎 = 𝐴 → ((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = ((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))))
109eqeq1d 2773 . . . . 5 (𝑎 = 𝐴 → (((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒) ↔ ((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒)))
1110ralbidv 3135 . . . 4 (𝑎 = 𝐴 → (∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝑎)) +𝑒 (𝑀‘(𝑒𝑎))) = (𝑀𝑒) ↔ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒)))
1211elrab 3515 . . 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 3364 . . . . . 6 (𝐴 ∈ 𝒫 𝑂𝐴 ∈ V)
1413a1i 11 . . . . 5 (𝜑 → (𝐴 ∈ 𝒫 𝑂𝐴 ∈ V))
15 simpr 471 . . . . . . 7 ((𝜑𝐴𝑂) → 𝐴𝑂)
161adantr 466 . . . . . . 7 ((𝜑𝐴𝑂) → 𝑂𝑉)
17 ssexg 4938 . . . . . . 7 ((𝐴𝑂𝑂𝑉) → 𝐴 ∈ V)
1815, 16, 17syl2anc 573 . . . . . 6 ((𝜑𝐴𝑂) → 𝐴 ∈ V)
1918ex 397 . . . . 5 (𝜑 → (𝐴𝑂𝐴 ∈ V))
20 elpwg 4305 . . . . . 6 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑂𝐴𝑂))
2120a1i 11 . . . . 5 (𝜑 → (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑂𝐴𝑂)))
2214, 19, 21pm5.21ndd 368 . . . 4 (𝜑 → (𝐴 ∈ 𝒫 𝑂𝐴𝑂))
2322anbi1d 615 . . 3 (𝜑 → ((𝐴 ∈ 𝒫 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒)) ↔ (𝐴𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒𝐴)) +𝑒 (𝑀‘(𝑒𝐴))) = (𝑀𝑒))))
2412, 23syl5bb 272 . 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))))
254, 24bitrd 268 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 196  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  cdif 3720  cin 3722  wss 3723  𝒫 cpw 4297  wf 6027  cfv 6031  (class class class)co 6793  0cc0 10138  +∞cpnf 10273   +𝑒 cxad 12149  [,]cicc 12383  toCaraSigaccarsg 30703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-carsg 30704
This theorem is referenced by:  baselcarsg  30708  0elcarsg  30709  difelcarsg  30712  inelcarsg  30713  carsgclctunlem1  30719  carsgclctunlem2  30721  carsgclctun  30723
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