Theorem disjresundif

Description: Lemma for ressucdifsn2 . (Contributed by Peter Mazsa, 24-Jul-2024)

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

Proof

Step	Hyp	Ref	Expression
1		resundi	⊢ ( 𝑅 ↾ ( 𝐴 ∪ 𝐵 ) ) = ( ( 𝑅 ↾ 𝐴 ) ∪ ( 𝑅 ↾ 𝐵 ) )
2	1	difeq1i	⊢ ( ( 𝑅 ↾ ( 𝐴 ∪ 𝐵 ) ) ∖ ( 𝑅 ↾ 𝐵 ) ) = ( ( ( 𝑅 ↾ 𝐴 ) ∪ ( 𝑅 ↾ 𝐵 ) ) ∖ ( 𝑅 ↾ 𝐵 ) )
3		difun2	⊢ ( ( ( 𝑅 ↾ 𝐴 ) ∪ ( 𝑅 ↾ 𝐵 ) ) ∖ ( 𝑅 ↾ 𝐵 ) ) = ( ( 𝑅 ↾ 𝐴 ) ∖ ( 𝑅 ↾ 𝐵 ) )
4	2 3	eqtri	⊢ ( ( 𝑅 ↾ ( 𝐴 ∪ 𝐵 ) ) ∖ ( 𝑅 ↾ 𝐵 ) ) = ( ( 𝑅 ↾ 𝐴 ) ∖ ( 𝑅 ↾ 𝐵 ) )
5		disjresdif	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( ( 𝑅 ↾ 𝐴 ) ∖ ( 𝑅 ↾ 𝐵 ) ) = ( 𝑅 ↾ 𝐴 ) )
6	4 5	eqtrid	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( ( 𝑅 ↾ ( 𝐴 ∪ 𝐵 ) ) ∖ ( 𝑅 ↾ 𝐵 ) ) = ( 𝑅 ↾ 𝐴 ) )