Step |
Hyp |
Ref |
Expression |
1 |
|
cnmpt21.j |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
2 |
|
cnmpt21.k |
⊢ ( 𝜑 → 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) |
3 |
|
cnmpt2c.l |
⊢ ( 𝜑 → 𝐿 ∈ ( TopOn ‘ 𝑍 ) ) |
4 |
|
cnmpt2c.p |
⊢ ( 𝜑 → 𝑃 ∈ 𝑍 ) |
5 |
|
eqidd |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → 𝑃 = 𝑃 ) |
6 |
5
|
mpompt |
⊢ ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ 𝑃 ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝑃 ) |
7 |
|
txtopon |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) → ( 𝐽 ×t 𝐾 ) ∈ ( TopOn ‘ ( 𝑋 × 𝑌 ) ) ) |
8 |
1 2 7
|
syl2anc |
⊢ ( 𝜑 → ( 𝐽 ×t 𝐾 ) ∈ ( TopOn ‘ ( 𝑋 × 𝑌 ) ) ) |
9 |
8 3 4
|
cnmptc |
⊢ ( 𝜑 → ( 𝑧 ∈ ( 𝑋 × 𝑌 ) ↦ 𝑃 ) ∈ ( ( 𝐽 ×t 𝐾 ) Cn 𝐿 ) ) |
10 |
6 9
|
eqeltrrid |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌 ↦ 𝑃 ) ∈ ( ( 𝐽 ×t 𝐾 ) Cn 𝐿 ) ) |