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Theorem caragenval 45507
Description: The sigma-algebra generated by an outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
caragenval (𝑂 ∈ OutMeas β†’ (CaraGenβ€˜π‘‚) = {𝑒 ∈ 𝒫 βˆͺ dom 𝑂 ∣ βˆ€π‘Ž ∈ 𝒫 βˆͺ dom 𝑂((π‘‚β€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘‚β€˜(π‘Ž βˆ– 𝑒))) = (π‘‚β€˜π‘Ž)})
Distinct variable group:   𝑂,π‘Ž,𝑒

Proof of Theorem caragenval
Dummy variable π‘œ is distinct from all other variables.
StepHypRef Expression
1 id 22 . 2 (𝑂 ∈ OutMeas β†’ 𝑂 ∈ OutMeas)
2 dmexg 7896 . . . . 5 (𝑂 ∈ OutMeas β†’ dom 𝑂 ∈ V)
32uniexd 7734 . . . 4 (𝑂 ∈ OutMeas β†’ βˆͺ dom 𝑂 ∈ V)
43pwexd 5376 . . 3 (𝑂 ∈ OutMeas β†’ 𝒫 βˆͺ dom 𝑂 ∈ V)
5 rabexg 5330 . . 3 (𝒫 βˆͺ dom 𝑂 ∈ V β†’ {𝑒 ∈ 𝒫 βˆͺ dom 𝑂 ∣ βˆ€π‘Ž ∈ 𝒫 βˆͺ dom 𝑂((π‘‚β€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘‚β€˜(π‘Ž βˆ– 𝑒))) = (π‘‚β€˜π‘Ž)} ∈ V)
64, 5syl 17 . 2 (𝑂 ∈ OutMeas β†’ {𝑒 ∈ 𝒫 βˆͺ dom 𝑂 ∣ βˆ€π‘Ž ∈ 𝒫 βˆͺ dom 𝑂((π‘‚β€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘‚β€˜(π‘Ž βˆ– 𝑒))) = (π‘‚β€˜π‘Ž)} ∈ V)
7 dmeq 5902 . . . . . 6 (π‘œ = 𝑂 β†’ dom π‘œ = dom 𝑂)
87unieqd 4921 . . . . 5 (π‘œ = 𝑂 β†’ βˆͺ dom π‘œ = βˆͺ dom 𝑂)
98pweqd 4618 . . . 4 (π‘œ = 𝑂 β†’ 𝒫 βˆͺ dom π‘œ = 𝒫 βˆͺ dom 𝑂)
109raleqdv 3323 . . . . 5 (π‘œ = 𝑂 β†’ (βˆ€π‘Ž ∈ 𝒫 βˆͺ dom π‘œ((π‘œβ€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘œβ€˜(π‘Ž βˆ– 𝑒))) = (π‘œβ€˜π‘Ž) ↔ βˆ€π‘Ž ∈ 𝒫 βˆͺ dom 𝑂((π‘œβ€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘œβ€˜(π‘Ž βˆ– 𝑒))) = (π‘œβ€˜π‘Ž)))
11 fveq1 6889 . . . . . . . 8 (π‘œ = 𝑂 β†’ (π‘œβ€˜(π‘Ž ∩ 𝑒)) = (π‘‚β€˜(π‘Ž ∩ 𝑒)))
12 fveq1 6889 . . . . . . . 8 (π‘œ = 𝑂 β†’ (π‘œβ€˜(π‘Ž βˆ– 𝑒)) = (π‘‚β€˜(π‘Ž βˆ– 𝑒)))
1311, 12oveq12d 7429 . . . . . . 7 (π‘œ = 𝑂 β†’ ((π‘œβ€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘œβ€˜(π‘Ž βˆ– 𝑒))) = ((π‘‚β€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘‚β€˜(π‘Ž βˆ– 𝑒))))
14 fveq1 6889 . . . . . . 7 (π‘œ = 𝑂 β†’ (π‘œβ€˜π‘Ž) = (π‘‚β€˜π‘Ž))
1513, 14eqeq12d 2746 . . . . . 6 (π‘œ = 𝑂 β†’ (((π‘œβ€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘œβ€˜(π‘Ž βˆ– 𝑒))) = (π‘œβ€˜π‘Ž) ↔ ((π‘‚β€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘‚β€˜(π‘Ž βˆ– 𝑒))) = (π‘‚β€˜π‘Ž)))
1615ralbidv 3175 . . . . 5 (π‘œ = 𝑂 β†’ (βˆ€π‘Ž ∈ 𝒫 βˆͺ dom 𝑂((π‘œβ€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘œβ€˜(π‘Ž βˆ– 𝑒))) = (π‘œβ€˜π‘Ž) ↔ βˆ€π‘Ž ∈ 𝒫 βˆͺ dom 𝑂((π‘‚β€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘‚β€˜(π‘Ž βˆ– 𝑒))) = (π‘‚β€˜π‘Ž)))
1710, 16bitrd 278 . . . 4 (π‘œ = 𝑂 β†’ (βˆ€π‘Ž ∈ 𝒫 βˆͺ dom π‘œ((π‘œβ€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘œβ€˜(π‘Ž βˆ– 𝑒))) = (π‘œβ€˜π‘Ž) ↔ βˆ€π‘Ž ∈ 𝒫 βˆͺ dom 𝑂((π‘‚β€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘‚β€˜(π‘Ž βˆ– 𝑒))) = (π‘‚β€˜π‘Ž)))
189, 17rabeqbidv 3447 . . 3 (π‘œ = 𝑂 β†’ {𝑒 ∈ 𝒫 βˆͺ dom π‘œ ∣ βˆ€π‘Ž ∈ 𝒫 βˆͺ dom π‘œ((π‘œβ€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘œβ€˜(π‘Ž βˆ– 𝑒))) = (π‘œβ€˜π‘Ž)} = {𝑒 ∈ 𝒫 βˆͺ dom 𝑂 ∣ βˆ€π‘Ž ∈ 𝒫 βˆͺ dom 𝑂((π‘‚β€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘‚β€˜(π‘Ž βˆ– 𝑒))) = (π‘‚β€˜π‘Ž)})
19 df-caragen 45506 . . 3 CaraGen = (π‘œ ∈ OutMeas ↦ {𝑒 ∈ 𝒫 βˆͺ dom π‘œ ∣ βˆ€π‘Ž ∈ 𝒫 βˆͺ dom π‘œ((π‘œβ€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘œβ€˜(π‘Ž βˆ– 𝑒))) = (π‘œβ€˜π‘Ž)})
2018, 19fvmptg 6995 . 2 ((𝑂 ∈ OutMeas ∧ {𝑒 ∈ 𝒫 βˆͺ dom 𝑂 ∣ βˆ€π‘Ž ∈ 𝒫 βˆͺ dom 𝑂((π‘‚β€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘‚β€˜(π‘Ž βˆ– 𝑒))) = (π‘‚β€˜π‘Ž)} ∈ V) β†’ (CaraGenβ€˜π‘‚) = {𝑒 ∈ 𝒫 βˆͺ dom 𝑂 ∣ βˆ€π‘Ž ∈ 𝒫 βˆͺ dom 𝑂((π‘‚β€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘‚β€˜(π‘Ž βˆ– 𝑒))) = (π‘‚β€˜π‘Ž)})
211, 6, 20syl2anc 582 1 (𝑂 ∈ OutMeas β†’ (CaraGenβ€˜π‘‚) = {𝑒 ∈ 𝒫 βˆͺ dom 𝑂 ∣ βˆ€π‘Ž ∈ 𝒫 βˆͺ dom 𝑂((π‘‚β€˜(π‘Ž ∩ 𝑒)) +𝑒 (π‘‚β€˜(π‘Ž βˆ– 𝑒))) = (π‘‚β€˜π‘Ž)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  {crab 3430  Vcvv 3472   βˆ– cdif 3944   ∩ cin 3946  π’« cpw 4601  βˆͺ cuni 4907  dom cdm 5675  β€˜cfv 6542  (class class class)co 7411   +𝑒 cxad 13094  OutMeascome 45503  CaraGenccaragen 45505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-caragen 45506
This theorem is referenced by:  caragenel  45509  caragenss  45518  caratheodory  45542
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