Theorem disjresdisj

Description: The intersection of restrictions to disjoint is the empty class. (Contributed by Peter Mazsa, 24-Jul-2024)

	Ref	Expression
Assertion	disjresdisj	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( ( 𝑅 ↾ 𝐴 ) ∩ ( 𝑅 ↾ 𝐵 ) ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		resindi	⊢ ( 𝑅 ↾ ( 𝐴 ∩ 𝐵 ) ) = ( ( 𝑅 ↾ 𝐴 ) ∩ ( 𝑅 ↾ 𝐵 ) )
2		disjresin	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( 𝑅 ↾ ( 𝐴 ∩ 𝐵 ) ) = ∅ )
3	1 2	eqtr3id	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( ( 𝑅 ↾ 𝐴 ) ∩ ( 𝑅 ↾ 𝐵 ) ) = ∅ )