Step |
Hyp |
Ref |
Expression |
1 |
|
cnmptid.j |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
2 |
|
cnmpt11.a |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ∈ ( 𝐽 Cn 𝐾 ) ) |
3 |
|
cnmpt1t.b |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) ∈ ( 𝐽 Cn 𝐿 ) ) |
4 |
|
cnmpt12f.f |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐾 ×t 𝐿 ) Cn 𝑀 ) ) |
5 |
|
df-ov |
⊢ ( 𝐴 𝐹 𝐵 ) = ( 𝐹 ‘ 〈 𝐴 , 𝐵 〉 ) |
6 |
5
|
mpteq2i |
⊢ ( 𝑥 ∈ 𝑋 ↦ ( 𝐴 𝐹 𝐵 ) ) = ( 𝑥 ∈ 𝑋 ↦ ( 𝐹 ‘ 〈 𝐴 , 𝐵 〉 ) ) |
7 |
1 2 3
|
cnmpt1t |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 〈 𝐴 , 𝐵 〉 ) ∈ ( 𝐽 Cn ( 𝐾 ×t 𝐿 ) ) ) |
8 |
1 7 4
|
cnmpt11f |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ ( 𝐹 ‘ 〈 𝐴 , 𝐵 〉 ) ) ∈ ( 𝐽 Cn 𝑀 ) ) |
9 |
6 8
|
eqeltrid |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ ( 𝐴 𝐹 𝐵 ) ) ∈ ( 𝐽 Cn 𝑀 ) ) |