Metamath Proof Explorer


Theorem resopunitintvd

Description: Restrict continuous function on open unit interval. (Contributed by metakunt, 12-May-2024)

Ref Expression
Hypothesis resopunitintvd.1 ( 𝜑 → ( 𝑥 ∈ ℂ ↦ 𝐴 ) ∈ ( ℂ –cn→ ℂ ) )
Assertion resopunitintvd ( 𝜑 → ( 𝑥 ∈ ( 0 (,) 1 ) ↦ 𝐴 ) ∈ ( ( 0 (,) 1 ) –cn→ ℂ ) )

Proof

Step Hyp Ref Expression
1 resopunitintvd.1 ( 𝜑 → ( 𝑥 ∈ ℂ ↦ 𝐴 ) ∈ ( ℂ –cn→ ℂ ) )
2 ioosscn ( 0 (,) 1 ) ⊆ ℂ
3 resmpt ( ( 0 (,) 1 ) ⊆ ℂ → ( ( 𝑥 ∈ ℂ ↦ 𝐴 ) ↾ ( 0 (,) 1 ) ) = ( 𝑥 ∈ ( 0 (,) 1 ) ↦ 𝐴 ) )
4 2 3 ax-mp ( ( 𝑥 ∈ ℂ ↦ 𝐴 ) ↾ ( 0 (,) 1 ) ) = ( 𝑥 ∈ ( 0 (,) 1 ) ↦ 𝐴 )
5 rescncf ( ( 0 (,) 1 ) ⊆ ℂ → ( ( 𝑥 ∈ ℂ ↦ 𝐴 ) ∈ ( ℂ –cn→ ℂ ) → ( ( 𝑥 ∈ ℂ ↦ 𝐴 ) ↾ ( 0 (,) 1 ) ) ∈ ( ( 0 (,) 1 ) –cn→ ℂ ) ) )
6 2 5 ax-mp ( ( 𝑥 ∈ ℂ ↦ 𝐴 ) ∈ ( ℂ –cn→ ℂ ) → ( ( 𝑥 ∈ ℂ ↦ 𝐴 ) ↾ ( 0 (,) 1 ) ) ∈ ( ( 0 (,) 1 ) –cn→ ℂ ) )
7 1 6 syl ( 𝜑 → ( ( 𝑥 ∈ ℂ ↦ 𝐴 ) ↾ ( 0 (,) 1 ) ) ∈ ( ( 0 (,) 1 ) –cn→ ℂ ) )
8 4 7 eqeltrrid ( 𝜑 → ( 𝑥 ∈ ( 0 (,) 1 ) ↦ 𝐴 ) ∈ ( ( 0 (,) 1 ) –cn→ ℂ ) )