Metamath Proof Explorer


Theorem snvonmbl

Description: A n-dimensional singleton is Lebesgue measurable. This is the first statement in Proposition 115G (e) of Fremlin1 p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021)

Ref Expression
Hypotheses snvonmbl.1 ( 𝜑𝑋 ∈ Fin )
snvonmbl.2 ( 𝜑𝐴 ∈ ( ℝ ↑m 𝑋 ) )
Assertion snvonmbl ( 𝜑 → { 𝐴 } ∈ dom ( voln ‘ 𝑋 ) )

Proof

Step Hyp Ref Expression
1 snvonmbl.1 ( 𝜑𝑋 ∈ Fin )
2 snvonmbl.2 ( 𝜑𝐴 ∈ ( ℝ ↑m 𝑋 ) )
3 2 rrxsnicc ( 𝜑X 𝑘𝑋 ( ( 𝐴𝑘 ) [,] ( 𝐴𝑘 ) ) = { 𝐴 } )
4 3 eqcomd ( 𝜑 → { 𝐴 } = X 𝑘𝑋 ( ( 𝐴𝑘 ) [,] ( 𝐴𝑘 ) ) )
5 eqid dom ( voln ‘ 𝑋 ) = dom ( voln ‘ 𝑋 )
6 elmapi ( 𝐴 ∈ ( ℝ ↑m 𝑋 ) → 𝐴 : 𝑋 ⟶ ℝ )
7 2 6 syl ( 𝜑𝐴 : 𝑋 ⟶ ℝ )
8 1 5 7 7 iccvonmbl ( 𝜑X 𝑘𝑋 ( ( 𝐴𝑘 ) [,] ( 𝐴𝑘 ) ) ∈ dom ( voln ‘ 𝑋 ) )
9 4 8 eqeltrd ( 𝜑 → { 𝐴 } ∈ dom ( voln ‘ 𝑋 ) )