Theorem disjresdif

Description: The difference between restrictions to disjoint is the first restriction. (Contributed by Peter Mazsa, 24-Jul-2024)

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

Proof

Step	Hyp	Ref	Expression
1		disjresdisj	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( ( 𝑅 ↾ 𝐴 ) ∩ ( 𝑅 ↾ 𝐵 ) ) = ∅ )
2		disjdif2	⊢ ( ( ( 𝑅 ↾ 𝐴 ) ∩ ( 𝑅 ↾ 𝐵 ) ) = ∅ → ( ( 𝑅 ↾ 𝐴 ) ∖ ( 𝑅 ↾ 𝐵 ) ) = ( 𝑅 ↾ 𝐴 ) )
3	1 2	syl	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( ( 𝑅 ↾ 𝐴 ) ∖ ( 𝑅 ↾ 𝐵 ) ) = ( 𝑅 ↾ 𝐴 ) )