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Theorem caragenval 46449
Description: The sigma-algebra generated by an outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
caragenval (𝑂 ∈ OutMeas → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
Distinct variable group:   𝑂,𝑎,𝑒

Proof of Theorem caragenval
Dummy variable 𝑜 is distinct from all other variables.
StepHypRef Expression
1 id 22 . 2 (𝑂 ∈ OutMeas → 𝑂 ∈ OutMeas)
2 dmexg 7924 . . . . 5 (𝑂 ∈ OutMeas → dom 𝑂 ∈ V)
32uniexd 7761 . . . 4 (𝑂 ∈ OutMeas → dom 𝑂 ∈ V)
43pwexd 5385 . . 3 (𝑂 ∈ OutMeas → 𝒫 dom 𝑂 ∈ V)
5 rabexg 5343 . . 3 (𝒫 dom 𝑂 ∈ V → {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)} ∈ V)
64, 5syl 17 . 2 (𝑂 ∈ OutMeas → {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)} ∈ V)
7 dmeq 5917 . . . . . 6 (𝑜 = 𝑂 → dom 𝑜 = dom 𝑂)
87unieqd 4925 . . . . 5 (𝑜 = 𝑂 dom 𝑜 = dom 𝑂)
98pweqd 4622 . . . 4 (𝑜 = 𝑂 → 𝒫 dom 𝑜 = 𝒫 dom 𝑂)
109raleqdv 3324 . . . . 5 (𝑜 = 𝑂 → (∀𝑎 ∈ 𝒫 dom 𝑜((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎) ↔ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎)))
11 fveq1 6906 . . . . . . . 8 (𝑜 = 𝑂 → (𝑜‘(𝑎𝑒)) = (𝑂‘(𝑎𝑒)))
12 fveq1 6906 . . . . . . . 8 (𝑜 = 𝑂 → (𝑜‘(𝑎𝑒)) = (𝑂‘(𝑎𝑒)))
1311, 12oveq12d 7449 . . . . . . 7 (𝑜 = 𝑂 → ((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = ((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))))
14 fveq1 6906 . . . . . . 7 (𝑜 = 𝑂 → (𝑜𝑎) = (𝑂𝑎))
1513, 14eqeq12d 2751 . . . . . 6 (𝑜 = 𝑂 → (((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎) ↔ ((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)))
1615ralbidv 3176 . . . . 5 (𝑜 = 𝑂 → (∀𝑎 ∈ 𝒫 dom 𝑂((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎) ↔ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)))
1710, 16bitrd 279 . . . 4 (𝑜 = 𝑂 → (∀𝑎 ∈ 𝒫 dom 𝑜((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎) ↔ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)))
189, 17rabeqbidv 3452 . . 3 (𝑜 = 𝑂 → {𝑒 ∈ 𝒫 dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 dom 𝑜((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎)} = {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
19 df-caragen 46448 . . 3 CaraGen = (𝑜 ∈ OutMeas ↦ {𝑒 ∈ 𝒫 dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 dom 𝑜((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎)})
2018, 19fvmptg 7014 . 2 ((𝑂 ∈ OutMeas ∧ {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)} ∈ V) → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
211, 6, 20syl2anc 584 1 (𝑂 ∈ OutMeas → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  cdif 3960  cin 3962  𝒫 cpw 4605   cuni 4912  dom cdm 5689  cfv 6563  (class class class)co 7431   +𝑒 cxad 13150  OutMeascome 46445  CaraGenccaragen 46447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-caragen 46448
This theorem is referenced by:  caragenel  46451  caragenss  46460  caratheodory  46484
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