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Theorem caragenval 44808
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 7845 . . . . 5 (𝑂 ∈ OutMeas β†’ dom 𝑂 ∈ V)
32uniexd 7684 . . . 4 (𝑂 ∈ OutMeas β†’ βˆͺ dom 𝑂 ∈ V)
43pwexd 5339 . . 3 (𝑂 ∈ OutMeas β†’ 𝒫 βˆͺ dom 𝑂 ∈ V)
5 rabexg 5293 . . 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 5864 . . . . . 6 (π‘œ = 𝑂 β†’ dom π‘œ = dom 𝑂)
87unieqd 4884 . . . . 5 (π‘œ = 𝑂 β†’ βˆͺ dom π‘œ = βˆͺ dom 𝑂)
98pweqd 4582 . . . 4 (π‘œ = 𝑂 β†’ 𝒫 βˆͺ dom π‘œ = 𝒫 βˆͺ dom 𝑂)
109raleqdv 3316 . . . . 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 6846 . . . . . . . 8 (π‘œ = 𝑂 β†’ (π‘œβ€˜(π‘Ž ∩ 𝑒)) = (π‘‚β€˜(π‘Ž ∩ 𝑒)))
12 fveq1 6846 . . . . . . . 8 (π‘œ = 𝑂 β†’ (π‘œβ€˜(π‘Ž βˆ– 𝑒)) = (π‘‚β€˜(π‘Ž βˆ– 𝑒)))
1311, 12oveq12d 7380 . . . . . . 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 6846 . . . . . . 7 (π‘œ = 𝑂 β†’ (π‘œβ€˜π‘Ž) = (π‘‚β€˜π‘Ž))
1513, 14eqeq12d 2753 . . . . . 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 279 . . . 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 3427 . . 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 44807 . . 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 6951 . 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 585 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 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3410  Vcvv 3448   βˆ– cdif 3912   ∩ cin 3914  π’« cpw 4565  βˆͺ cuni 4870  dom cdm 5638  β€˜cfv 6501  (class class class)co 7362   +𝑒 cxad 13038  OutMeascome 44804  CaraGenccaragen 44806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-caragen 44807
This theorem is referenced by:  caragenel  44810  caragenss  44819  caratheodory  44843
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