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