Theorem caragensspw 46524

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

Hypotheses

Ref	Expression
caragensspw.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
caragensspw.x	⊢ 𝑋 = ∪ dom 𝑂
caragensspw.s	⊢ 𝑆 = (CaraGen‘𝑂)

Assertion

Ref	Expression
caragensspw	⊢ (𝜑 → 𝑆 ⊆ 𝒫 𝑋)

Proof of Theorem caragensspw

Step	Hyp	Ref	Expression
1		caragensspw.o	. . . 4 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
2		caragensspw.s	. . . . 5 ⊢ 𝑆 = (CaraGen‘𝑂)
3	2	caragenss 46519	. . . 4 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
4	1, 3	syl 17	. . 3 ⊢ (𝜑 → 𝑆 ⊆ dom 𝑂)
5		pwuni 4945	. . . 4 ⊢ dom 𝑂 ⊆ 𝒫 ∪ dom 𝑂
6	5	a1i 11	. . 3 ⊢ (𝜑 → dom 𝑂 ⊆ 𝒫 ∪ dom 𝑂)
7	4, 6	sstrd 3994	. 2 ⊢ (𝜑 → 𝑆 ⊆ 𝒫 ∪ dom 𝑂)
8		caragensspw.x	. . . . 5 ⊢ 𝑋 = ∪ dom 𝑂
9	8	pweqi 4616	. . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
10	9	eqcomi 2746	. . 3 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋
11	10	a1i 11	. 2 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋)
12	7, 11	sseqtrd 4020	1 ⊢ (𝜑 → 𝑆 ⊆ 𝒫 𝑋)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951  𝒫 cpw 4600  ∪ cuni 4907  dom cdm 5685  ‘cfv 6561  OutMeascome 46504  CaraGenccaragen 46506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-ome 46505  df-caragen 46507

This theorem is referenced by:  caratheodorylem2  46542