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Metamath Proof Explorer
Description: Closure law for an operation. (Contributed by NM, 19-Apr-2007) (Proof
shortened by AV, 9-Mar-2025)
|
|
Ref |
Expression |
|
Hypothesis |
fovcl.1 |
⊢ 𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 |
|
Assertion |
fovcl |
⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 𝐹 𝐵 ) ∈ 𝐶 ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
fovcl.1 |
⊢ 𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 |
| 2 |
1
|
a1i |
⊢ ( 𝐴 ∈ 𝑅 → 𝐹 : ( 𝑅 × 𝑆 ) ⟶ 𝐶 ) |
| 3 |
2
|
fovcld |
⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 𝐹 𝐵 ) ∈ 𝐶 ) |
| 4 |
3
|
3anidm12 |
⊢ ( ( 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 𝐹 𝐵 ) ∈ 𝐶 ) |