Metamath Proof Explorer


Theorem elhoi

Description: Membership in a multidimensional half-open interval. (Contributed by Glauco Siliprandi, 11-Oct-2020)

Ref Expression
Hypothesis elhoi.1 ( 𝜑𝑋𝑉 )
Assertion elhoi ( 𝜑 → ( 𝑌 ∈ ( ( 𝐴 [,) 𝐵 ) ↑m 𝑋 ) ↔ ( 𝑌 : 𝑋 ⟶ ℝ* ∧ ∀ 𝑥𝑋 ( 𝑌𝑥 ) ∈ ( 𝐴 [,) 𝐵 ) ) ) )

Proof

Step Hyp Ref Expression
1 elhoi.1 ( 𝜑𝑋𝑉 )
2 ovexd ( 𝜑 → ( 𝐴 [,) 𝐵 ) ∈ V )
3 elmapg ( ( ( 𝐴 [,) 𝐵 ) ∈ V ∧ 𝑋𝑉 ) → ( 𝑌 ∈ ( ( 𝐴 [,) 𝐵 ) ↑m 𝑋 ) ↔ 𝑌 : 𝑋 ⟶ ( 𝐴 [,) 𝐵 ) ) )
4 2 1 3 syl2anc ( 𝜑 → ( 𝑌 ∈ ( ( 𝐴 [,) 𝐵 ) ↑m 𝑋 ) ↔ 𝑌 : 𝑋 ⟶ ( 𝐴 [,) 𝐵 ) ) )
5 id ( 𝑌 : 𝑋 ⟶ ( 𝐴 [,) 𝐵 ) → 𝑌 : 𝑋 ⟶ ( 𝐴 [,) 𝐵 ) )
6 icossxr ( 𝐴 [,) 𝐵 ) ⊆ ℝ*
7 6 a1i ( 𝑌 : 𝑋 ⟶ ( 𝐴 [,) 𝐵 ) → ( 𝐴 [,) 𝐵 ) ⊆ ℝ* )
8 5 7 fssd ( 𝑌 : 𝑋 ⟶ ( 𝐴 [,) 𝐵 ) → 𝑌 : 𝑋 ⟶ ℝ* )
9 ffvelrn ( ( 𝑌 : 𝑋 ⟶ ( 𝐴 [,) 𝐵 ) ∧ 𝑥𝑋 ) → ( 𝑌𝑥 ) ∈ ( 𝐴 [,) 𝐵 ) )
10 9 ralrimiva ( 𝑌 : 𝑋 ⟶ ( 𝐴 [,) 𝐵 ) → ∀ 𝑥𝑋 ( 𝑌𝑥 ) ∈ ( 𝐴 [,) 𝐵 ) )
11 8 10 jca ( 𝑌 : 𝑋 ⟶ ( 𝐴 [,) 𝐵 ) → ( 𝑌 : 𝑋 ⟶ ℝ* ∧ ∀ 𝑥𝑋 ( 𝑌𝑥 ) ∈ ( 𝐴 [,) 𝐵 ) ) )
12 ffn ( 𝑌 : 𝑋 ⟶ ℝ*𝑌 Fn 𝑋 )
13 12 adantr ( ( 𝑌 : 𝑋 ⟶ ℝ* ∧ ∀ 𝑥𝑋 ( 𝑌𝑥 ) ∈ ( 𝐴 [,) 𝐵 ) ) → 𝑌 Fn 𝑋 )
14 simpr ( ( 𝑌 : 𝑋 ⟶ ℝ* ∧ ∀ 𝑥𝑋 ( 𝑌𝑥 ) ∈ ( 𝐴 [,) 𝐵 ) ) → ∀ 𝑥𝑋 ( 𝑌𝑥 ) ∈ ( 𝐴 [,) 𝐵 ) )
15 13 14 jca ( ( 𝑌 : 𝑋 ⟶ ℝ* ∧ ∀ 𝑥𝑋 ( 𝑌𝑥 ) ∈ ( 𝐴 [,) 𝐵 ) ) → ( 𝑌 Fn 𝑋 ∧ ∀ 𝑥𝑋 ( 𝑌𝑥 ) ∈ ( 𝐴 [,) 𝐵 ) ) )
16 ffnfv ( 𝑌 : 𝑋 ⟶ ( 𝐴 [,) 𝐵 ) ↔ ( 𝑌 Fn 𝑋 ∧ ∀ 𝑥𝑋 ( 𝑌𝑥 ) ∈ ( 𝐴 [,) 𝐵 ) ) )
17 15 16 sylibr ( ( 𝑌 : 𝑋 ⟶ ℝ* ∧ ∀ 𝑥𝑋 ( 𝑌𝑥 ) ∈ ( 𝐴 [,) 𝐵 ) ) → 𝑌 : 𝑋 ⟶ ( 𝐴 [,) 𝐵 ) )
18 11 17 impbii ( 𝑌 : 𝑋 ⟶ ( 𝐴 [,) 𝐵 ) ↔ ( 𝑌 : 𝑋 ⟶ ℝ* ∧ ∀ 𝑥𝑋 ( 𝑌𝑥 ) ∈ ( 𝐴 [,) 𝐵 ) ) )
19 18 a1i ( 𝜑 → ( 𝑌 : 𝑋 ⟶ ( 𝐴 [,) 𝐵 ) ↔ ( 𝑌 : 𝑋 ⟶ ℝ* ∧ ∀ 𝑥𝑋 ( 𝑌𝑥 ) ∈ ( 𝐴 [,) 𝐵 ) ) ) )
20 4 19 bitrd ( 𝜑 → ( 𝑌 ∈ ( ( 𝐴 [,) 𝐵 ) ↑m 𝑋 ) ↔ ( 𝑌 : 𝑋 ⟶ ℝ* ∧ ∀ 𝑥𝑋 ( 𝑌𝑥 ) ∈ ( 𝐴 [,) 𝐵 ) ) ) )