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Metamath Proof Explorer

Theorem ioof

Description: The set of open intervals of extended reals maps to subsets of reals. (Contributed by NM, 7-Feb-2007) (Revised by Mario Carneiro, 16-Nov-2013)

		Ref	Expression
	Assertion	ioof	⊢ (,) : ( ℝ^* × ℝ^* ) ⟶ 𝒫 ℝ

Proof

Step	Hyp	Ref	Expression
1		iooval	⊢ ( ( 𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ) → ( 𝑥 (,) 𝑦 ) = { 𝑧 ∈ ℝ^* ∣ ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) } )
2		ioossre	⊢ ( 𝑥 (,) 𝑦 ) ⊆ ℝ
3		ovex	⊢ ( 𝑥 (,) 𝑦 ) ∈ V
4	3	elpw	⊢ ( ( 𝑥 (,) 𝑦 ) ∈ 𝒫 ℝ ↔ ( 𝑥 (,) 𝑦 ) ⊆ ℝ )
5	2 4	mpbir	⊢ ( 𝑥 (,) 𝑦 ) ∈ 𝒫 ℝ
6	1 5	eqeltrrdi	⊢ ( ( 𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ) → { 𝑧 ∈ ℝ^* ∣ ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) } ∈ 𝒫 ℝ )
7	6	rgen2	⊢ ∀ 𝑥 ∈ ℝ^* ∀ 𝑦 ∈ ℝ^* { 𝑧 ∈ ℝ^* ∣ ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) } ∈ 𝒫 ℝ
8		df-ioo	⊢ (,) = ( 𝑥 ∈ ℝ^* , 𝑦 ∈ ℝ^* ↦ { 𝑧 ∈ ℝ^* ∣ ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) } )
9	8	fmpo	⊢ ( ∀ 𝑥 ∈ ℝ^* ∀ 𝑦 ∈ ℝ^* { 𝑧 ∈ ℝ^* ∣ ( 𝑥 < 𝑧 ∧ 𝑧 < 𝑦 ) } ∈ 𝒫 ℝ ↔ (,) : ( ℝ^* × ℝ^* ) ⟶ 𝒫 ℝ )
10	7 9	mpbi	⊢ (,) : ( ℝ^* × ℝ^* ) ⟶ 𝒫 ℝ