Theorem hoissre

Description: The projection of a half-open interval onto a single dimension is a subset of RR . (Contributed by Glauco Siliprandi, 11-Oct-2020)

	Ref	Expression
Hypothesis	hoissre.1	⊢ ( 𝜑 → 𝐼 : 𝑋 ⟶ ( ℝ × ℝ ) )
Assertion	hoissre	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ 𝐼 ) ‘ 𝑘 ) ⊆ ℝ )

Proof

Step	Hyp	Ref	Expression
1		hoissre.1	⊢ ( 𝜑 → 𝐼 : 𝑋 ⟶ ( ℝ × ℝ ) )
2	1	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝐼 : 𝑋 ⟶ ( ℝ × ℝ ) )
3		simpr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝑘 ∈ 𝑋 )
4	2 3	fvovco	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ 𝐼 ) ‘ 𝑘 ) = ( ( 1st ‘ ( 𝐼 ‘ 𝑘 ) ) [,) ( 2nd ‘ ( 𝐼 ‘ 𝑘 ) ) ) )
5	1	ffvelrnda	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 𝐼 ‘ 𝑘 ) ∈ ( ℝ × ℝ ) )
6		xp1st	⊢ ( ( 𝐼 ‘ 𝑘 ) ∈ ( ℝ × ℝ ) → ( 1st ‘ ( 𝐼 ‘ 𝑘 ) ) ∈ ℝ )
7	5 6	syl	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 1st ‘ ( 𝐼 ‘ 𝑘 ) ) ∈ ℝ )
8		xp2nd	⊢ ( ( 𝐼 ‘ 𝑘 ) ∈ ( ℝ × ℝ ) → ( 2nd ‘ ( 𝐼 ‘ 𝑘 ) ) ∈ ℝ )
9	5 8	syl	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 2nd ‘ ( 𝐼 ‘ 𝑘 ) ) ∈ ℝ )
10	9	rexrd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 2nd ‘ ( 𝐼 ‘ 𝑘 ) ) ∈ ℝ* )
11		icossre	⊢ ( ( ( 1st ‘ ( 𝐼 ‘ 𝑘 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝐼 ‘ 𝑘 ) ) ∈ ℝ* ) → ( ( 1st ‘ ( 𝐼 ‘ 𝑘 ) ) [,) ( 2nd ‘ ( 𝐼 ‘ 𝑘 ) ) ) ⊆ ℝ )
12	7 10 11	syl2anc	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( ( 1st ‘ ( 𝐼 ‘ 𝑘 ) ) [,) ( 2nd ‘ ( 𝐼 ‘ 𝑘 ) ) ) ⊆ ℝ )
13	4 12	eqsstrd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( ( [,) ∘ 𝐼 ) ‘ 𝑘 ) ⊆ ℝ )