Theorem resabs1d

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 ) )