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Theorem caragenval 43721
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 7690 . . . . 5 (𝑂 ∈ OutMeas → dom 𝑂 ∈ V)
32uniexd 7539 . . . 4 (𝑂 ∈ OutMeas → dom 𝑂 ∈ V)
43pwexd 5281 . . 3 (𝑂 ∈ OutMeas → 𝒫 dom 𝑂 ∈ V)
5 rabexg 5233 . . 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 5781 . . . . . 6 (𝑜 = 𝑂 → dom 𝑜 = dom 𝑂)
87unieqd 4842 . . . . 5 (𝑜 = 𝑂 dom 𝑜 = dom 𝑂)
98pweqd 4541 . . . 4 (𝑜 = 𝑂 → 𝒫 dom 𝑜 = 𝒫 dom 𝑂)
109raleqdv 3332 . . . . 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 6725 . . . . . . . 8 (𝑜 = 𝑂 → (𝑜‘(𝑎𝑒)) = (𝑂‘(𝑎𝑒)))
12 fveq1 6725 . . . . . . . 8 (𝑜 = 𝑂 → (𝑜‘(𝑎𝑒)) = (𝑂‘(𝑎𝑒)))
1311, 12oveq12d 7240 . . . . . . 7 (𝑜 = 𝑂 → ((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = ((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))))
14 fveq1 6725 . . . . . . 7 (𝑜 = 𝑂 → (𝑜𝑎) = (𝑂𝑎))
1513, 14eqeq12d 2754 . . . . . 6 (𝑜 = 𝑂 → (((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎) ↔ ((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)))
1615ralbidv 3119 . . . . 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 3403 . . 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 43720 . . 3 CaraGen = (𝑜 ∈ OutMeas ↦ {𝑒 ∈ 𝒫 dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 dom 𝑜((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎)})
2018, 19fvmptg 6825 . 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 1543  wcel 2111  wral 3062  {crab 3066  Vcvv 3415  cdif 3872  cin 3874  𝒫 cpw 4522   cuni 4828  dom cdm 5560  cfv 6389  (class class class)co 7222   +𝑒 cxad 12715  OutMeascome 43717  CaraGenccaragen 43719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5201  ax-nul 5208  ax-pow 5267  ax-pr 5331  ax-un 7532
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3417  df-dif 3878  df-un 3880  df-in 3882  df-ss 3892  df-nul 4247  df-if 4449  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4829  df-br 5063  df-opab 5125  df-mpt 5145  df-id 5464  df-xp 5566  df-rel 5567  df-cnv 5568  df-co 5569  df-dm 5570  df-rn 5571  df-iota 6347  df-fun 6391  df-fv 6397  df-ov 7225  df-caragen 43720
This theorem is referenced by:  caragenel  43723  caragenss  43732  caratheodory  43756
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