Step |
Hyp |
Ref |
Expression |
1 |
|
fovcld.1 |
⊢ ( 𝜑 → 𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 ) |
2 |
|
3simpc |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) ) |
3 |
|
ffnov |
⊢ ( 𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 ↔ ( 𝐹 Fn ( 𝑅 × 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑅 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 ) ) |
4 |
3
|
simprbi |
⊢ ( 𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 → ∀ 𝑥 ∈ 𝑅 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 ) |
5 |
1 4
|
syl |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑅 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 ) |
6 |
5
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ∀ 𝑥 ∈ 𝑅 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 ) |
7 |
|
oveq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐹 𝑦 ) = ( 𝐴 𝐹 𝑦 ) ) |
8 |
7
|
eleq1d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 ↔ ( 𝐴 𝐹 𝑦 ) ∈ 𝐶 ) ) |
9 |
|
oveq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐵 ) ) |
10 |
9
|
eleq1d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝐹 𝑦 ) ∈ 𝐶 ↔ ( 𝐴 𝐹 𝐵 ) ∈ 𝐶 ) ) |
11 |
8 10
|
rspc2v |
⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( ∀ 𝑥 ∈ 𝑅 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐹 𝑦 ) ∈ 𝐶 → ( 𝐴 𝐹 𝐵 ) ∈ 𝐶 ) ) |
12 |
2 6 11
|
sylc |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 𝐹 𝐵 ) ∈ 𝐶 ) |