Theorem areass

Description: A measurable region is a subset of RR X. RR . (Contributed by Mario Carneiro, 21-Jun-2015)

	Ref	Expression
Assertion	areass	⊢ ( 𝑆 ∈ dom area → 𝑆 ⊆ ( ℝ × ℝ ) )

Proof

Step	Hyp	Ref	Expression
1		dmarea	⊢ ( 𝑆 ∈ dom area ↔ ( 𝑆 ⊆ ( ℝ × ℝ ) ∧ ∀ 𝑥 ∈ ℝ ( 𝑆 “ { 𝑥 } ) ∈ ( ◡ vol “ ℝ ) ∧ ( 𝑥 ∈ ℝ ↦ ( vol ‘ ( 𝑆 “ { 𝑥 } ) ) ) ∈ 𝐿1 ) )
2	1	simp1bi	⊢ ( 𝑆 ∈ dom area → 𝑆 ⊆ ( ℝ × ℝ ) )