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 ) ) |