Theorem resabs2

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

	Ref	Expression
Assertion	resabs2	⊢ ( 𝐵 ⊆ 𝐶 → ( ( 𝐴 ↾ 𝐵 ) ↾ 𝐶 ) = ( 𝐴 ↾ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		rescom	⊢ ( ( 𝐴 ↾ 𝐵 ) ↾ 𝐶 ) = ( ( 𝐴 ↾ 𝐶 ) ↾ 𝐵 )
2		resabs1	⊢ ( 𝐵 ⊆ 𝐶 → ( ( 𝐴 ↾ 𝐶 ) ↾ 𝐵 ) = ( 𝐴 ↾ 𝐵 ) )
3	1 2	eqtrid	⊢ ( 𝐵 ⊆ 𝐶 → ( ( 𝐴 ↾ 𝐵 ) ↾ 𝐶 ) = ( 𝐴 ↾ 𝐵 ) )