Theorem disjiminres

Description: Disjointness condition for intersection with restriction. (Contributed by Peter Mazsa, 27-Sep-2021)

	Ref	Expression
Assertion	disjiminres	⊢ ( Disj 𝑆 → Disj ( 𝑅 ∩ ( 𝑆 ↾ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		disjimres	⊢ ( Disj 𝑆 → Disj ( 𝑆 ↾ 𝐴 ) )
2		disjimin	⊢ ( Disj ( 𝑆 ↾ 𝐴 ) → Disj ( 𝑅 ∩ ( 𝑆 ↾ 𝐴 ) ) )
3	1 2	syl	⊢ ( Disj 𝑆 → Disj ( 𝑅 ∩ ( 𝑆 ↾ 𝐴 ) ) )