Metamath Proof Explorer


Theorem resabs1d

Description: Absorption law for restriction, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypothesis resabs1d.b ( 𝜑𝐵𝐶 )
Assertion resabs1d ( 𝜑 → ( ( 𝐴𝐶 ) ↾ 𝐵 ) = ( 𝐴𝐵 ) )

Proof

Step Hyp Ref Expression
1 resabs1d.b ( 𝜑𝐵𝐶 )
2 resabs1 ( 𝐵𝐶 → ( ( 𝐴𝐶 ) ↾ 𝐵 ) = ( 𝐴𝐵 ) )
3 1 2 syl ( 𝜑 → ( ( 𝐴𝐶 ) ↾ 𝐵 ) = ( 𝐴𝐵 ) )