Metamath Proof Explorer
Description: Absorption law for restriction, deduction form. (Contributed by Glauco
Siliprandi, 11-Dec-2019)
|
|
Ref |
Expression |
|
Hypothesis |
resabs1d.b |
⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 ) |
|
Assertion |
resabs1d |
⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝐶 ) ↾ 𝐵 ) = ( 𝐴 ↾ 𝐵 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
resabs1d.b |
⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 ) |
2 |
|
resabs1 |
⊢ ( 𝐵 ⊆ 𝐶 → ( ( 𝐴 ↾ 𝐶 ) ↾ 𝐵 ) = ( 𝐴 ↾ 𝐵 ) ) |
3 |
1 2
|
syl |
⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝐶 ) ↾ 𝐵 ) = ( 𝐴 ↾ 𝐵 ) ) |