Theorem unidmovn

Description: Base set of the n-dimensional Lebesgue outer measure. (Contributed by Glauco Siliprandi, 24-Dec-2020)

	Ref	Expression
Hypothesis	unidmovn.1	⊢ ( 𝜑 → 𝑋 ∈ Fin )
Assertion	unidmovn	⊢ ( 𝜑 → ∪ dom ( voln* ‘ 𝑋 ) = ( ℝ ↑m 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		unidmovn.1	⊢ ( 𝜑 → 𝑋 ∈ Fin )
2	1	dmovn	⊢ ( 𝜑 → dom ( voln* ‘ 𝑋 ) = 𝒫 ( ℝ ↑m 𝑋 ) )
3	2	unieqd	⊢ ( 𝜑 → ∪ dom ( voln* ‘ 𝑋 ) = ∪ 𝒫 ( ℝ ↑m 𝑋 ) )
4		unipw	⊢ ∪ 𝒫 ( ℝ ↑m 𝑋 ) = ( ℝ ↑m 𝑋 )
5	4	a1i	⊢ ( 𝜑 → ∪ 𝒫 ( ℝ ↑m 𝑋 ) = ( ℝ ↑m 𝑋 ) )
6	3 5	eqtrd	⊢ ( 𝜑 → ∪ dom ( voln* ‘ 𝑋 ) = ( ℝ ↑m 𝑋 ) )