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Theorem caragenval 41494
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 7358 . . . . 5 (𝑂 ∈ OutMeas → dom 𝑂 ∈ V)
3 uniexg 7215 . . . . 5 (dom 𝑂 ∈ V → dom 𝑂 ∈ V)
42, 3syl 17 . . . 4 (𝑂 ∈ OutMeas → dom 𝑂 ∈ V)
54pwexd 5079 . . 3 (𝑂 ∈ OutMeas → 𝒫 dom 𝑂 ∈ V)
6 rabexg 5036 . . 3 (𝒫 dom 𝑂 ∈ V → {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)} ∈ V)
75, 6syl 17 . 2 (𝑂 ∈ OutMeas → {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)} ∈ V)
8 dmeq 5556 . . . . . 6 (𝑜 = 𝑂 → dom 𝑜 = dom 𝑂)
98unieqd 4668 . . . . 5 (𝑜 = 𝑂 dom 𝑜 = dom 𝑂)
109pweqd 4383 . . . 4 (𝑜 = 𝑂 → 𝒫 dom 𝑜 = 𝒫 dom 𝑂)
1110raleqdv 3356 . . . . 5 (𝑜 = 𝑂 → (∀𝑎 ∈ 𝒫 dom 𝑜((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎) ↔ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎)))
12 fveq1 6432 . . . . . . . 8 (𝑜 = 𝑂 → (𝑜‘(𝑎𝑒)) = (𝑂‘(𝑎𝑒)))
13 fveq1 6432 . . . . . . . 8 (𝑜 = 𝑂 → (𝑜‘(𝑎𝑒)) = (𝑂‘(𝑎𝑒)))
1412, 13oveq12d 6923 . . . . . . 7 (𝑜 = 𝑂 → ((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = ((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))))
15 fveq1 6432 . . . . . . 7 (𝑜 = 𝑂 → (𝑜𝑎) = (𝑂𝑎))
1614, 15eqeq12d 2840 . . . . . 6 (𝑜 = 𝑂 → (((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎) ↔ ((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)))
1716ralbidv 3195 . . . . 5 (𝑜 = 𝑂 → (∀𝑎 ∈ 𝒫 dom 𝑂((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎) ↔ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)))
1811, 17bitrd 271 . . . 4 (𝑜 = 𝑂 → (∀𝑎 ∈ 𝒫 dom 𝑜((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎) ↔ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)))
1910, 18rabeqbidv 3408 . . 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)})
20 df-caragen 41493 . . 3 CaraGen = (𝑜 ∈ OutMeas ↦ {𝑒 ∈ 𝒫 dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 dom 𝑜((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎)})
2119, 20fvmptg 6527 . 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)})
221, 7, 21syl2anc 579 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 1656  wcel 2164  wral 3117  {crab 3121  Vcvv 3414  cdif 3795  cin 3797  𝒫 cpw 4378   cuni 4658  dom cdm 5342  cfv 6123  (class class class)co 6905   +𝑒 cxad 12230  OutMeascome 41490  CaraGenccaragen 41492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fv 6131  df-ov 6908  df-caragen 41493
This theorem is referenced by:  caragenel  41496  caragenss  41505  caratheodory  41529
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