Theorem caragensspw 46941

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 46936	. . . 4 ⊢ (𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂)
4	1, 3	syl 17	. . 3 ⊢ (𝜑 → 𝑆 ⊆ dom 𝑂)
5		pwuni 4889	. . . 4 ⊢ dom 𝑂 ⊆ 𝒫 ∪ dom 𝑂
6	5	a1i 11	. . 3 ⊢ (𝜑 → dom 𝑂 ⊆ 𝒫 ∪ dom 𝑂)
7	4, 6	sstrd 3933	. 2 ⊢ (𝜑 → 𝑆 ⊆ 𝒫 ∪ dom 𝑂)
8		caragensspw.x	. . . . 5 ⊢ 𝑋 = ∪ dom 𝑂
9	8	pweqi 4558	. . . 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 3959	1 ⊢ (𝜑 → 𝑆 ⊆ 𝒫 𝑋)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851  dom cdm 5622  ‘cfv 6490  OutMeascome 46921  CaraGenccaragen 46923

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 2185  ax-ext 2709  ax-sep 5231  ax-pow 5300  ax-pr 5368  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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-ov 7361  df-ome 46922  df-caragen 46924

This theorem is referenced by:  caratheodorylem2  46959