Theorem caragenval 46414

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 7941	. . . . 5 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 ∈ V)
3	2	uniexd 7777	. . . 4 ⊢ (𝑂 ∈ OutMeas → ∪ dom 𝑂 ∈ V)
4	3	pwexd 5397	. . 3 ⊢ (𝑂 ∈ OutMeas → 𝒫 ∪ dom 𝑂 ∈ V)
5		rabexg 5355	. . 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 5928	. . . . . 6 ⊢ (𝑜 = 𝑂 → dom 𝑜 = dom 𝑂)
8	7	unieqd 4944	. . . . 5 ⊢ (𝑜 = 𝑂 → ∪ dom 𝑜 = ∪ dom 𝑂)
9	8	pweqd 4639	. . . 4 ⊢ (𝑜 = 𝑂 → 𝒫 ∪ dom 𝑜 = 𝒫 ∪ dom 𝑂)
10	9	raleqdv 3334	. . . . 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 6919	. . . . . . . 8 ⊢ (𝑜 = 𝑂 → (𝑜‘(𝑎 ∩ 𝑒)) = (𝑂‘(𝑎 ∩ 𝑒)))
12		fveq1 6919	. . . . . . . 8 ⊢ (𝑜 = 𝑂 → (𝑜‘(𝑎 ∖ 𝑒)) = (𝑂‘(𝑎 ∖ 𝑒)))
13	11, 12	oveq12d 7466	. . . . . . 7 ⊢ (𝑜 = 𝑂 → ((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = ((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))))
14		fveq1 6919	. . . . . . 7 ⊢ (𝑜 = 𝑂 → (𝑜‘𝑎) = (𝑂‘𝑎))
15	13, 14	eqeq12d 2756	. . . . . 6 ⊢ (𝑜 = 𝑂 → (((𝑜‘(𝑎 ∩ 𝑒)) +𝑒 (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎) ↔ ((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)))
16	15	ralbidv 3184	. . . . 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 3462	. . 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 46413	. . 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 7027	. 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 583	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 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443  Vcvv 3488   ∖ cdif 3973   ∩ cin 3975  𝒫 cpw 4622  ∪ cuni 4931  dom cdm 5700  ‘cfv 6573  (class class class)co 7448   +_𝑒 cxad 13173  OutMeascome 46410  CaraGenccaragen 46412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-caragen 46413

This theorem is referenced by:  caragenel  46416  caragenss  46425  caratheodory  46449