Theorem caragenval 46480

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‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)})

Distinct variable group:   𝑂,𝑎,𝑒

Proof of Theorem caragenval

Dummy variable 𝑜 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝑂 ∈ OutMeas → 𝑂 ∈ OutMeas)
2		dmexg 7905	. . . . 5 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 ∈ V)
3	2	uniexd 7744	. . . 4 ⊢ (𝑂 ∈ OutMeas → ∪ dom 𝑂 ∈ V)
4	3	pwexd 5359	. . 3 ⊢ (𝑂 ∈ OutMeas → 𝒫 ∪ dom 𝑂 ∈ V)
5		rabexg 5317	. . 3 ⊢ (𝒫 ∪ dom 𝑂 ∈ V → {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ∈ V)
6	4, 5	syl 17	. 2 ⊢ (𝑂 ∈ OutMeas → {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ∈ V)
7		dmeq 5894	. . . . . 6 ⊢ (𝑜 = 𝑂 → dom 𝑜 = dom 𝑂)
8	7	unieqd 4900	. . . . 5 ⊢ (𝑜 = 𝑂 → ∪ dom 𝑜 = ∪ dom 𝑂)
9	8	pweqd 4597	. . . 4 ⊢ (𝑜 = 𝑂 → 𝒫 ∪ dom 𝑜 = 𝒫 ∪ dom 𝑂)
10	9	raleqdv 3309	. . . . 5 ⊢ (𝑜 = 𝑂 → (∀𝑎 ∈ 𝒫 ∪ dom 𝑜((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎) ↔ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎)))
11		fveq1 6885	. . . . . . . 8 ⊢ (𝑜 = 𝑂 → (𝑜‘(𝑎 ∩ 𝑒)) = (𝑂‘(𝑎 ∩ 𝑒)))
12		fveq1 6885	. . . . . . . 8 ⊢ (𝑜 = 𝑂 → (𝑜‘(𝑎 ∖ 𝑒)) = (𝑂‘(𝑎 ∖ 𝑒)))
13	11, 12	oveq12d 7431	. . . . . . 7 ⊢ (𝑜 = 𝑂 → ((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = ((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))))
14		fveq1 6885	. . . . . . 7 ⊢ (𝑜 = 𝑂 → (𝑜‘𝑎) = (𝑂‘𝑎))
15	13, 14	eqeq12d 2750	. . . . . 6 ⊢ (𝑜 = 𝑂 → (((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎) ↔ ((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)))
16	15	ralbidv 3165	. . . . 5 ⊢ (𝑜 = 𝑂 → (∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎) ↔ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)))
17	10, 16	bitrd 279	. . . 4 ⊢ (𝑜 = 𝑂 → (∀𝑎 ∈ 𝒫 ∪ dom 𝑜((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎) ↔ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)))
18	9, 17	rabeqbidv 3438	. . 3 ⊢ (𝑜 = 𝑂 → {𝑒 ∈ 𝒫 ∪ dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑜((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎)} = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)})
19		df-caragen 46479	. . 3 ⊢ CaraGen = (𝑜 ∈ OutMeas ↦ {𝑒 ∈ 𝒫 ∪ dom 𝑜 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑜((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎)})
20	18, 19	fvmptg 6994	. 2 ⊢ ((𝑂 ∈ OutMeas ∧ {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ∈ V) → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)})
21	1, 6, 20	syl2anc 584	1 ⊢ (𝑂 ∈ OutMeas → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ∀wral 3050  {crab 3419  Vcvv 3463   ∖ cdif 3928   ∩ cin 3930  𝒫 cpw 4580  ∪ cuni 4887  dom cdm 5665  ‘cfv 6541  (class class class)co 7413   +_𝑒 cxad 13134  OutMeascome 46476  CaraGenccaragen 46478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-iota 6494  df-fun 6543  df-fv 6549  df-ov 7416  df-caragen 46479

This theorem is referenced by:  caragenel  46482  caragenss  46491  caratheodory  46515