Metamath Proof Explorer
Description: Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014)
|
|
Ref |
Expression |
|
Hypotheses |
ofmresex.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
|
|
ofmresex.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
|
Assertion |
ofmresex |
⊢ ( 𝜑 → ( ∘f 𝑅 ↾ ( 𝐴 × 𝐵 ) ) ∈ V ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
ofmresex.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
ofmresex.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
3 |
1 2
|
xpexd |
⊢ ( 𝜑 → ( 𝐴 × 𝐵 ) ∈ V ) |
4 |
|
ofexg |
⊢ ( ( 𝐴 × 𝐵 ) ∈ V → ( ∘f 𝑅 ↾ ( 𝐴 × 𝐵 ) ) ∈ V ) |
5 |
3 4
|
syl |
⊢ ( 𝜑 → ( ∘f 𝑅 ↾ ( 𝐴 × 𝐵 ) ) ∈ V ) |