Metamath Proof Explorer


Theorem vonmblss2

Description: n-dimensional Lebesgue measurable sets are subsets of the n-dimensional real Euclidean space. (Contributed by Glauco Siliprandi, 3-Mar-2021)

Ref Expression
Hypotheses vonmblss2.x ( 𝜑𝑋 ∈ Fin )
vonmblss2.y ( 𝜑𝑌 ∈ dom ( voln ‘ 𝑋 ) )
Assertion vonmblss2 ( 𝜑𝑌 ⊆ ( ℝ ↑m 𝑋 ) )

Proof

Step Hyp Ref Expression
1 vonmblss2.x ( 𝜑𝑋 ∈ Fin )
2 vonmblss2.y ( 𝜑𝑌 ∈ dom ( voln ‘ 𝑋 ) )
3 1 vonmblss ( 𝜑 → dom ( voln ‘ 𝑋 ) ⊆ 𝒫 ( ℝ ↑m 𝑋 ) )
4 3 2 sseldd ( 𝜑𝑌 ∈ 𝒫 ( ℝ ↑m 𝑋 ) )
5 elpwi ( 𝑌 ∈ 𝒫 ( ℝ ↑m 𝑋 ) → 𝑌 ⊆ ( ℝ ↑m 𝑋 ) )
6 4 5 syl ( 𝜑𝑌 ⊆ ( ℝ ↑m 𝑋 ) )