Theorem caragenss 47069

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 4034	. . 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 47058	. . . 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 2798	. . 3 ⊢ (𝑂 ∈ OutMeas → 𝑆 = {𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝑒)) +𝑒 (𝑂‘(𝑎 ∖ 𝑒))) = (𝑂‘𝑎)})
7		omedm 47064	. . 3 ⊢ (𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 ∪ dom 𝑂)
8	6, 7	sseq12d 3970	. 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 259	1 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)

Syntax hints:   → wi 4   = wceq 1561   ∈ wcel 2143  ∀wral 3077  {crab 3415   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4556  ∪ cuni 4866  dom cdm 5648  ‘cfv 6521  (class class class)co 7396   +_𝑒 cxad 13122  OutMeascome 47054  CaraGenccaragen 47056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-pow 5323  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-ov 7399  df-ome 47055  df-caragen 47057

This theorem is referenced by:  caragensspw  47074  caragenuni  47076  caragendifcl  47079  caratheodorylem1  47091  caratheodorylem2  47092  dmvon  47171  voncmpl  47186  vonmblss  47205