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Metamath Proof Explorer

Theorem dfioo2

Description: Alternate definition of the set of open intervals of extended reals. (Contributed by NM, 1-Mar-2007) (Revised by Mario Carneiro, 1-Sep-2015)

		Ref	Expression
	Assertion	dfioo2	⊢ (,) = ( 𝑥 ∈ ℝ^* , 𝑦 ∈ ℝ^* ↦ { 𝑤 ∈ ℝ ∣ ( 𝑥 < 𝑤 ∧ 𝑤 < 𝑦 ) } )

Proof

Step	Hyp	Ref	Expression
1		ioof	⊢ (,) : ( ℝ^* × ℝ^* ) ⟶ 𝒫 ℝ
2		ffn	⊢ ( (,) : ( ℝ^* × ℝ^* ) ⟶ 𝒫 ℝ → (,) Fn ( ℝ^* × ℝ^* ) )
3	1 2	ax-mp	⊢ (,) Fn ( ℝ^* × ℝ^* )
4		fnov	⊢ ( (,) Fn ( ℝ^* × ℝ^* ) ↔ (,) = ( 𝑥 ∈ ℝ^* , 𝑦 ∈ ℝ^* ↦ ( 𝑥 (,) 𝑦 ) ) )
5	3 4	mpbi	⊢ (,) = ( 𝑥 ∈ ℝ^* , 𝑦 ∈ ℝ^* ↦ ( 𝑥 (,) 𝑦 ) )
6		iooval2	⊢ ( ( 𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^* ) → ( 𝑥 (,) 𝑦 ) = { 𝑤 ∈ ℝ ∣ ( 𝑥 < 𝑤 ∧ 𝑤 < 𝑦 ) } )
7	6	mpoeq3ia	⊢ ( 𝑥 ∈ ℝ^* , 𝑦 ∈ ℝ^* ↦ ( 𝑥 (,) 𝑦 ) ) = ( 𝑥 ∈ ℝ^* , 𝑦 ∈ ℝ^* ↦ { 𝑤 ∈ ℝ ∣ ( 𝑥 < 𝑤 ∧ 𝑤 < 𝑦 ) } )
8	5 7	eqtri	⊢ (,) = ( 𝑥 ∈ ℝ^* , 𝑦 ∈ ℝ^* ↦ { 𝑤 ∈ ℝ ∣ ( 𝑥 < 𝑤 ∧ 𝑤 < 𝑦 ) } )