Theorem caragensspw 47324

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 47319	. . . 4 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
4	1, 3	syl 18	. . 3 ⊢ (𝜑 → 𝑆 ⊆ dom 𝑂)
5		pwuni 4909	. . . 4 ⊢ dom 𝑂 ⊆ 𝒫 ∪ dom 𝑂
6	5	a1i 11	. . 3 ⊢ (𝜑 → dom 𝑂 ⊆ 𝒫 ∪ dom 𝑂)
7	4, 6	sstrd 3944	. 2 ⊢ (𝜑 → 𝑆 ⊆ 𝒫 ∪ dom 𝑂)
8		caragensspw.x	. . . . 5 ⊢ 𝑋 = ∪ dom 𝑂
9	8	pweqi 4576	. . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
10	9	eqcomi 2771	. . 3 ⊢ 𝒫 ∪ dom 𝑂 = 𝒫 𝑋
11	10	a1i 11	. 2 ⊢ (𝜑 → 𝒫 ∪ dom 𝑂 = 𝒫 𝑋)
12	7, 11	sseqtrd 3970	1 ⊢ (𝜑 → 𝑆 ⊆ 𝒫 𝑋)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ⊆ wss 3902  𝒫 cpw 4560  ∪ cuni 4870  dom cdm 5659  ‘cfv 6537  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:  caratheodorylem2  47342