Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
2 |
|
eqid |
⊢ ∪ 𝐾 = ∪ 𝐾 |
3 |
1 2
|
cnpf |
⊢ ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → 𝐹 : ∪ 𝐽 ⟶ ∪ 𝐾 ) |
4 |
|
toponuni |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 ) |
5 |
4
|
feq2d |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → ( 𝐹 : 𝑋 ⟶ 𝑌 ↔ 𝐹 : ∪ 𝐽 ⟶ 𝑌 ) ) |
6 |
|
toponuni |
⊢ ( 𝐾 ∈ ( TopOn ‘ 𝑌 ) → 𝑌 = ∪ 𝐾 ) |
7 |
6
|
feq3d |
⊢ ( 𝐾 ∈ ( TopOn ‘ 𝑌 ) → ( 𝐹 : ∪ 𝐽 ⟶ 𝑌 ↔ 𝐹 : ∪ 𝐽 ⟶ ∪ 𝐾 ) ) |
8 |
5 7
|
sylan9bb |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) → ( 𝐹 : 𝑋 ⟶ 𝑌 ↔ 𝐹 : ∪ 𝐽 ⟶ ∪ 𝐾 ) ) |
9 |
3 8
|
syl5ibr |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) → ( 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) → 𝐹 : 𝑋 ⟶ 𝑌 ) ) |
10 |
9
|
3impia |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐹 ∈ ( ( 𝐽 CnP 𝐾 ) ‘ 𝑃 ) ) → 𝐹 : 𝑋 ⟶ 𝑌 ) |