Step |
Hyp |
Ref |
Expression |
1 |
|
iscn.1 |
⊢ 𝑋 = ∪ 𝐽 |
2 |
|
iscn.2 |
⊢ 𝑌 = ∪ 𝐾 |
3 |
|
df-cn |
⊢ Cn = ( 𝑗 ∈ Top , 𝑘 ∈ Top ↦ { 𝑓 ∈ ( ∪ 𝑘 ↑m ∪ 𝑗 ) ∣ ∀ 𝑦 ∈ 𝑘 ( ◡ 𝑓 “ 𝑦 ) ∈ 𝑗 } ) |
4 |
3
|
elmpocl |
⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → ( 𝐽 ∈ Top ∧ 𝐾 ∈ Top ) ) |
5 |
1
|
toptopon |
⊢ ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
6 |
2
|
toptopon |
⊢ ( 𝐾 ∈ Top ↔ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) |
7 |
|
iscn |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) → ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑦 ∈ 𝐾 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ) ) |
8 |
5 6 7
|
syl2anb |
⊢ ( ( 𝐽 ∈ Top ∧ 𝐾 ∈ Top ) → ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑦 ∈ 𝐾 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ) ) |
9 |
4 8
|
biadanii |
⊢ ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ↔ ( ( 𝐽 ∈ Top ∧ 𝐾 ∈ Top ) ∧ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑦 ∈ 𝐾 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ) ) |