Theorem caragenss

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)

	Ref	Expression
Hypothesis	caragenss.1	⊢ 𝑆 = ( CaraGen ‘ 𝑂 )
Assertion	caragenss	⊢ ( 𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂 )

Proof

Step	Hyp	Ref	Expression
1		caragenss.1	⊢ 𝑆 = ( CaraGen ‘ 𝑂 )
2		ssrab2	⊢ { 𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀ 𝑎 ∈ 𝒫 ∪ dom 𝑂 ( ( 𝑂 ‘ ( 𝑎 ∩ 𝑒 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎 ∖ 𝑒 ) ) ) = ( 𝑂 ‘ 𝑎 ) } ⊆ 𝒫 ∪ dom 𝑂
3	2	a1i	⊢ ( 𝑂 ∈ OutMeas → { 𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀ 𝑎 ∈ 𝒫 ∪ dom 𝑂 ( ( 𝑂 ‘ ( 𝑎 ∩ 𝑒 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎 ∖ 𝑒 ) ) ) = ( 𝑂 ‘ 𝑎 ) } ⊆ 𝒫 ∪ dom 𝑂 )
4	1	a1i	⊢ ( 𝑂 ∈ OutMeas → 𝑆 = ( CaraGen ‘ 𝑂 ) )
5		caragenval	⊢ ( 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	⊢ ( 𝑂 ∈ OutMeas → 𝑆 = { 𝑒 ∈ 𝒫 ∪ dom 𝑂 ∣ ∀ 𝑎 ∈ 𝒫 ∪ dom 𝑂 ( ( 𝑂 ‘ ( 𝑎 ∩ 𝑒 ) ) +𝑒 ( 𝑂 ‘ ( 𝑎 ∖ 𝑒 ) ) ) = ( 𝑂 ‘ 𝑎 ) } )
7		omedm	⊢ ( 𝑂 ∈ OutMeas → dom 𝑂 = 𝒫 ∪ dom 𝑂 )
8	6 7	sseq12d	⊢ ( 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	3 8	mpbird	⊢ ( 𝑂 ∈ OutMeas → 𝑆 ⊆ dom 𝑂 )