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Theorem caragenval 40044
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 7059 . . . . 5 (𝑂 ∈ OutMeas → dom 𝑂 ∈ V)
3 uniexg 6920 . . . . 5 (dom 𝑂 ∈ V → dom 𝑂 ∈ V)
42, 3syl 17 . . . 4 (𝑂 ∈ OutMeas → dom 𝑂 ∈ V)
5 pwexg 4820 . . . 4 ( dom 𝑂 ∈ V → 𝒫 dom 𝑂 ∈ V)
64, 5syl 17 . . 3 (𝑂 ∈ OutMeas → 𝒫 dom 𝑂 ∈ V)
7 rabexg 4782 . . 3 (𝒫 dom 𝑂 ∈ V → {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)} ∈ V)
86, 7syl 17 . 2 (𝑂 ∈ OutMeas → {𝑒 ∈ 𝒫 dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)} ∈ V)
9 dmeq 5294 . . . . . 6 (𝑜 = 𝑂 → dom 𝑜 = dom 𝑂)
109unieqd 4419 . . . . 5 (𝑜 = 𝑂 dom 𝑜 = dom 𝑂)
1110pweqd 4141 . . . 4 (𝑜 = 𝑂 → 𝒫 dom 𝑜 = 𝒫 dom 𝑂)
1211raleqdv 3137 . . . . 5 (𝑜 = 𝑂 → (∀𝑎 ∈ 𝒫 dom 𝑜((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎) ↔ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎)))
13 fveq1 6157 . . . . . . . 8 (𝑜 = 𝑂 → (𝑜‘(𝑎𝑒)) = (𝑂‘(𝑎𝑒)))
14 fveq1 6157 . . . . . . . 8 (𝑜 = 𝑂 → (𝑜‘(𝑎𝑒)) = (𝑂‘(𝑎𝑒)))
1513, 14oveq12d 6633 . . . . . . 7 (𝑜 = 𝑂 → ((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = ((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))))
16 fveq1 6157 . . . . . . 7 (𝑜 = 𝑂 → (𝑜𝑎) = (𝑂𝑎))
1715, 16eqeq12d 2636 . . . . . 6 (𝑜 = 𝑂 → (((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎) ↔ ((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)))
1817ralbidv 2982 . . . . 5 (𝑜 = 𝑂 → (∀𝑎 ∈ 𝒫 dom 𝑂((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎) ↔ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)))
1912, 18bitrd 268 . . . 4 (𝑜 = 𝑂 → (∀𝑎 ∈ 𝒫 dom 𝑜((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎) ↔ ∀𝑎 ∈ 𝒫 dom 𝑂((𝑂‘(𝑎𝑒)) +𝑒 (𝑂‘(𝑎𝑒))) = (𝑂𝑎)))
2011, 19rabeqbidv 3185 . . 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)})
21 df-caragen 40043 . . 3 CaraGen = (𝑜 ∈ OutMeas ↦ {𝑒 ∈ 𝒫 dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 dom 𝑜((𝑜‘(𝑎𝑒)) +𝑒 (𝑜‘(𝑎𝑒))) = (𝑜𝑎)})
2220, 21fvmptg 6247 . 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)})
231, 8, 22syl2anc 692 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 1480  wcel 1987  wral 2908  {crab 2912  Vcvv 3190  cdif 3557  cin 3559  𝒫 cpw 4136   cuni 4409  dom cdm 5084  cfv 5857  (class class class)co 6615   +𝑒 cxad 11904  OutMeascome 40040  CaraGenccaragen 40042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-caragen 40043
This theorem is referenced by:  caragenel  40046  caragenss  40055  caratheodory  40079
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