Theorem disjresin

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

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

Proof

Step	Hyp	Ref	Expression
1		reseq2	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( 𝑅 ↾ ( 𝐴 ∩ 𝐵 ) ) = ( 𝑅 ↾ ∅ ) )
2		res0	⊢ ( 𝑅 ↾ ∅ ) = ∅
3	1 2	eqtrdi	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( 𝑅 ↾ ( 𝐴 ∩ 𝐵 ) ) = ∅ )