Theorem resabs1i

Description: Absorption law for restriction. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Hypothesis	resabs1i.1	⊢ 𝐵 ⊆ 𝐶
Assertion	resabs1i	⊢ ( ( 𝐴 ↾ 𝐶 ) ↾ 𝐵 ) = ( 𝐴 ↾ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		resabs1i.1	⊢ 𝐵 ⊆ 𝐶
2		resabs1	⊢ ( 𝐵 ⊆ 𝐶 → ( ( 𝐴 ↾ 𝐶 ) ↾ 𝐵 ) = ( 𝐴 ↾ 𝐵 ) )
3	1 2	ax-mp	⊢ ( ( 𝐴 ↾ 𝐶 ) ↾ 𝐵 ) = ( 𝐴 ↾ 𝐵 )