Theorem caragenss 46460

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 4090	. . 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 46449	. . . 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 2775	. . 3 ⊢ (𝑂 ∈ OutMeas → 𝑆 = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)})
7		omedm 46455	. . 3 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 ∪ dom 𝑂)
8	6, 7	sseq12d 4029	. 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 257	1 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433   ∖ cdif 3960   ∩ cin 3962   ⊆ wss 3963  𝒫 cpw 4605  ∪ cuni 4912  dom cdm 5689  ‘cfv 6563  (class class class)co 7431   +_𝑒 cxad 13150  OutMeascome 46445  CaraGenccaragen 46447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-ome 46446  df-caragen 46448

This theorem is referenced by:  caragensspw  46465  caragenuni  46467  caragendifcl  46470  caratheodorylem1  46482  caratheodorylem2  46483  dmvon  46562  voncmpl  46577  vonmblss  46596