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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	$⊢ (.) : ℝ^{} \times ℝ^{} ⟶ 𝒫 ℝ$

Proof

Step	Hyp	Ref	Expression
1		iooval	$⊢ (x \in ℝ^{} \land y \in ℝ^{}) \to (x, y) = \{z \in ℝ^{*} \| (x < z \land z < y)\}$
2		ioossre	$⊢ (x, y) \subseteq ℝ$
3		ovex	$⊢ (x, y) \in V$
4	3	elpw	$⊢ (x, y) \in 𝒫 ℝ \leftrightarrow (x, y) \subseteq ℝ$
5	2 4	mpbir	$⊢ (x, y) \in 𝒫 ℝ$
6	1 5	eqeltrrdi	$⊢ (x \in ℝ^{} \land y \in ℝ^{}) \to \{z \in ℝ^{*} \| (x < z \land z < y)\} \in 𝒫 ℝ$
7	6	rgen2	$⊢ \forall x \in ℝ^{} \forall y \in ℝ^{} \{z \in ℝ^{*} \| (x < z \land z < y)\} \in 𝒫 ℝ$
8		df-ioo	$⊢ (.) = (x \in ℝ^{}, y \in ℝ^{} ⟼ \{z \in ℝ^{*} \| (x < z \land z < y)\})$
9	8	fmpo	$⊢ \forall x \in ℝ^{} \forall y \in ℝ^{} \{z \in ℝ^{} \| (x < z \land z < y)\} \in 𝒫 ℝ \leftrightarrow (.) : ℝ^{} \times ℝ^{*} ⟶ 𝒫 ℝ$
10	7 9	mpbi	$⊢ (.) : ℝ^{} \times ℝ^{} ⟶ 𝒫 ℝ$