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Theorem caragenval 42923
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 7588 . . . . 5 (𝑂 ∈ OutMeas → dom 𝑂 ∈ V)
32uniexd 7443 . . . 4 (𝑂 ∈ OutMeas → dom 𝑂 ∈ V)
43pwexd 5253 . . 3 (𝑂 ∈ OutMeas → 𝒫 dom 𝑂 ∈ V)
5 rabexg 5207 . . 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 5745 . . . . . 6 (𝑜 = 𝑂 → dom 𝑜 = dom 𝑂)
87unieqd 4825 . . . . 5 (𝑜 = 𝑂 dom 𝑜 = dom 𝑂)
98pweqd 4531 . . . 4 (𝑜 = 𝑂 → 𝒫 dom 𝑜 = 𝒫 dom 𝑂)
109raleqdv 3396 . . . . 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 6642 . . . . . . . 8 (𝑜 = 𝑂 → (𝑜‘(𝑎𝑒)) = (𝑂‘(𝑎𝑒)))
12 fveq1 6642 . . . . . . . 8 (𝑜 = 𝑂 → (𝑜‘(𝑎𝑒)) = (𝑂‘(𝑎𝑒)))
1311, 12oveq12d 7148 . . . . . . 7 (𝑜 = 𝑂 → ((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = ((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))))
14 fveq1 6642 . . . . . . 7 (𝑜 = 𝑂 → (𝑜𝑎) = (𝑂𝑎))
1513, 14eqeq12d 2837 . . . . . 6 (𝑜 = 𝑂 → (((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎) ↔ ((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)))
1615ralbidv 3185 . . . . 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 282 . . . 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 3462 . . 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 42922 . . 3 CaraGen = (𝑜 ∈ OutMeas ↦ {𝑒 ∈ 𝒫 dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 dom 𝑜((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎)})
2018, 19fvmptg 6739 . 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 587 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 1538  wcel 2115  wral 3126  {crab 3130  Vcvv 3471  cdif 3907  cin 3909  𝒫 cpw 4512   cuni 4811  dom cdm 5528  cfv 6328  (class class class)co 7130   +𝑒 cxad 12483  OutMeascome 42919  CaraGenccaragen 42921
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7133  df-caragen 42922
This theorem is referenced by:  caragenel  42925  caragenss  42934  caratheodory  42958
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