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Metamath Proof Explorer

Theorem ixxf

Description: The set of intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007) (Revised by Mario Carneiro, 16-Nov-2013)

		Ref	Expression
	Hypothesis	ixx.1	$⊢ O = (x \in ℝ^{}, y \in ℝ^{} ⟼ \{z \in ℝ^{*} \| (x R z \land z S y)\})$
	Assertion	ixxf	$⊢ O : ℝ^{} \times ℝ^{} ⟶ 𝒫 ℝ^{*}$

Proof

Step	Hyp	Ref	Expression
1		ixx.1	$⊢ O = (x \in ℝ^{}, y \in ℝ^{} ⟼ \{z \in ℝ^{*} \| (x R z \land z S y)\})$
2		xrex	$⊢ ℝ^{*} \in V$
3		ssrab2	$⊢ \{z \in ℝ^{} \| (x R z \land z S y)\} \subseteq ℝ^{}$
4	2 3	elpwi2	$⊢ \{z \in ℝ^{} \| (x R z \land z S y)\} \in 𝒫 ℝ^{}$
5	4	rgen2w	$⊢ \forall x \in ℝ^{} \forall y \in ℝ^{} \{z \in ℝ^{} \| (x R z \land z S y)\} \in 𝒫 ℝ^{}$
6	1	fmpo	$⊢ \forall x \in ℝ^{} \forall y \in ℝ^{} \{z \in ℝ^{} \| (x R z \land z S y)\} \in 𝒫 ℝ^{} \leftrightarrow O : ℝ^{} \times ℝ^{} ⟶ 𝒫 ℝ^{*}$
7	5 6	mpbi	$⊢ O : ℝ^{} \times ℝ^{} ⟶ 𝒫 ℝ^{*}$