Theorem caragenss 46785

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 46774	. . . 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 2770	. . 3 ⊢ (𝑂 ∈ OutMeas → 𝑆 = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)})
7		omedm 46780	. . 3 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 ∪ dom 𝑂)
8	6, 7	sseq12d 3966	. 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 1542   ∈ wcel 2114  ∀wral 3050  {crab 3398   ∖ cdif 3897   ∩ cin 3899   ⊆ wss 3900  𝒫 cpw 4553  ∪ cuni 4862  dom cdm 5623  ‘cfv 6491  (class class class)co 7358   +_𝑒 cxad 13026  OutMeascome 46770  CaraGenccaragen 46772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-fv 6499  df-ov 7361  df-ome 46771  df-caragen 46773

This theorem is referenced by:  caragensspw  46790  caragenuni  46792  caragendifcl  46795  caratheodorylem1  46807  caratheodorylem2  46808  dmvon  46887  voncmpl  46902  vonmblss  46921