Theorem resabs2d

Description: Absorption law for restriction. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypothesis	resabs2d.1	⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
Assertion	resabs2d	⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝐵 ) ↾ 𝐶 ) = ( 𝐴 ↾ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		resabs2d.1	⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )
2		resabs2	⊢ ( 𝐵 ⊆ 𝐶 → ( ( 𝐴 ↾ 𝐵 ) ↾ 𝐶 ) = ( 𝐴 ↾ 𝐵 ) )
3	1 2	syl	⊢ ( 𝜑 → ( ( 𝐴 ↾ 𝐵 ) ↾ 𝐶 ) = ( 𝐴 ↾ 𝐵 ) )