Metamath Proof Explorer
Description: Restriction of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019)
|
|
Ref |
Expression |
|
Hypothesis |
resmptd.b |
⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 ) |
|
Assertion |
resmptd |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ↾ 𝐵 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
resmptd.b |
⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 ) |
2 |
|
resmpt |
⊢ ( 𝐵 ⊆ 𝐴 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ↾ 𝐵 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) ) |
3 |
1 2
|
syl |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ↾ 𝐵 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) ) |