Theorem caragenss 47319

Description: The sigma-algebra generated from an outer measure, by the Caratheodory's construction, is a subset of the domain of the outer measure. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
caragenss.1	⊢ 𝑆 = (CaraGen‘𝑂)

Assertion

Ref	Expression
caragenss	⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)

Proof of Theorem caragenss

Dummy variables 𝑎 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4031	. . 3 ⊢ {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ⊆ 𝒫 ∪ dom 𝑂
2	1	a1i 11	. 2 ⊢ (𝑂 ∈ OutMeas → {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ⊆ 𝒫 ∪ dom 𝑂)
3		caragenss.1	. . . . 5 ⊢ 𝑆 = (CaraGen‘𝑂)
4	3	a1i 11	. . . 4 ⊢ (𝑂 ∈ OutMeas → 𝑆 = (CaraGen‘𝑂))
5		caragenval 47308	. . . 4 ⊢ (𝑂 ∈ OutMeas → (CaraGen‘𝑂) = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)})
6	4, 5	eqtrd 2797	. . 3 ⊢ (𝑂 ∈ OutMeas → 𝑆 = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)})
7		omedm 47314	. . 3 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 ∪ dom 𝑂)
8	6, 7	sseq12d 3967	. 2 ⊢ (𝑂 ∈ OutMeas → (𝑆 ⊆ dom 𝑂 ↔ {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)} ⊆ 𝒫 ∪ dom 𝑂))
9	2, 8	mpbird 260	1 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4560  ∪ cuni 4870  dom cdm 5659  ‘cfv 6537  (class class class)co 7416   +_𝑒 cxad 13163  OutMeascome 47304  CaraGenccaragen 47306

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7419  df-ome 47305  df-caragen 47307

This theorem is used by:  caragensspw  47324  caragenuni  47326  caragendifcl  47329  caratheodorylem1  47341  caratheodorylem2  47342  dmvon  47421  voncmpl  47436  vonmblss  47455