Metamath Proof Explorer


Theorem resabs2

Description: Absorption law for restriction. (Contributed by NM, 27-Mar-1998)

Ref Expression
Assertion resabs2 ( 𝐵𝐶 → ( ( 𝐴𝐵 ) ↾ 𝐶 ) = ( 𝐴𝐵 ) )

Proof

Step Hyp Ref Expression
1 rescom ( ( 𝐴𝐵 ) ↾ 𝐶 ) = ( ( 𝐴𝐶 ) ↾ 𝐵 )
2 resabs1 ( 𝐵𝐶 → ( ( 𝐴𝐶 ) ↾ 𝐵 ) = ( 𝐴𝐵 ) )
3 1 2 eqtrid ( 𝐵𝐶 → ( ( 𝐴𝐵 ) ↾ 𝐶 ) = ( 𝐴𝐵 ) )