Metamath Proof Explorer
Description: A continuous function at point P is a mapping. (Contributed by FL, 17-Nov-2006) (Revised by Mario Carneiro, 21-Aug-2015)
|
|
Ref |
Expression |
|
Hypotheses |
iscnp2.1 |
⊢ 𝑋 = ∪ 𝐽 |
|
|
iscnp2.2 |
⊢ 𝑌 = ∪ 𝐾 |
|
Assertion |
cnpf |
⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → 𝐹 : 𝑋 ⟶ 𝑌 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
iscnp2.1 |
⊢ 𝑋 = ∪ 𝐽 |
2 |
|
iscnp2.2 |
⊢ 𝑌 = ∪ 𝐾 |
3 |
1 2
|
iscnp2 |
⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ↔ ( ( 𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑃 ∈ 𝑋 ) ∧ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑦 ∈ 𝐾 ( ( 𝐹 ‘ 𝑃 ) ∈ 𝑦 → ∃ 𝑥 ∈ 𝐽 ( 𝑃 ∈ 𝑥 ∧ ( 𝐹 “ 𝑥 ) ⊆ 𝑦 ) ) ) ) ) |
4 |
3
|
simprbi |
⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑦 ∈ 𝐾 ( ( 𝐹 ‘ 𝑃 ) ∈ 𝑦 → ∃ 𝑥 ∈ 𝐽 ( 𝑃 ∈ 𝑥 ∧ ( 𝐹 “ 𝑥 ) ⊆ 𝑦 ) ) ) ) |
5 |
4
|
simpld |
⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → 𝐹 : 𝑋 ⟶ 𝑌 ) |