Theorem caragenval 47546

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 23	. 2 ⊢ (𝑂 ∈ OutMeas → 𝑂 ∈ OutMeas)
2		dmexg 7913	. . . . 5 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 ∈ V)
3	2	uniexd 7759	. . . 4 ⊢ (𝑂 ∈ OutMeas → ∪ dom 𝑂 ∈ V)
4	3	pwexd 5340	. . 3 ⊢ (𝑂 ∈ OutMeas → 𝒫 ∪ dom 𝑂 ∈ V)
5		rabexg 5298	. . 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 18	. 2 ⊢ (𝑂 ∈ OutMeas → {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +e (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ∈ V)
7		dmeq 5884	. . . . . 6 ⊢ (𝑜 = 𝑂 → dom 𝑜 = dom 𝑂)
8	7	unieqd 4879	. . . . 5 ⊢ (𝑜 = 𝑂 → ∪ dom 𝑜 = ∪ dom 𝑂)
9	8	pweqd 4573	. . . 4 ⊢ (𝑜 = 𝑂 → 𝒫 ∪ dom 𝑜 = 𝒫 ∪ dom 𝑂)
10	9	raleqdv 3319	. . . . 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 6884	. . . . . . . 8 ⊢ (𝑜 = 𝑂 → (𝑜‘(𝑎 ∩ 𝑒)) = (𝑂‘(𝑎 ∩ 𝑒)))
12		fveq1 6884	. . . . . . . 8 ⊢ (𝑜 = 𝑂 → (𝑜‘(𝑎 ∖ 𝑒)) = (𝑂‘(𝑎 ∖ 𝑒)))
13	11, 12	oveq12d 7438	. . . . . . 7 ⊢ (𝑜 = 𝑂 → ((𝑜‘(𝑎 ∩ 𝑒)) +e (𝑜‘(𝑎 ∖ 𝑒))) = ((𝑂‘(𝑎 ∩ 𝑒)) +e (𝑂‘(𝑎 ∖ 𝑒))))
14		fveq1 6884	. . . . . . 7 ⊢ (𝑜 = 𝑂 → (𝑜‘𝑎) = (𝑂‘𝑎))
15	13, 14	eqeq12d 2776	. . . . . 6 ⊢ (𝑜 = 𝑂 → (((𝑜‘(𝑎 ∩ 𝑒)) +e (𝑜‘(𝑎 ∖ 𝑒))) = (𝑜‘𝑎) ↔ ((𝑂‘(𝑎 ∩ 𝑒)) +e (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)))
16	15	ralbidv 3185	. . . . 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 282	. . . 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 3429	. . 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 47545	. . 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 6991	. 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 596	1 ⊢ (𝑂 ∈ OutMeas → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +e (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)})

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412  Vcvv 3450   ∖ cdif 4045   ∩ cin 4089  𝒫 cpw 4556  ∪ cuni 4866  dom cdm 5650  ‘cfv 6538  (class class class)co 7420   +_e cxad 13239  OutMeascome 47542  CaraGenccaragen 47544

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-caragen 47545

This theorem is used by:  caragenel  47548  caragenss  47557  caratheodory  47581