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 i^i B ) = (/) -> ( ( R |` A ) i^i ( R |` B ) ) = (/) )

Proof

Step	Hyp	Ref	Expression
1		resindi	|- ( R |` ( A i^i B ) ) = ( ( R |` A ) i^i ( R |` B ) )
2		disjresin	|- ( ( A i^i B ) = (/) -> ( R |` ( A i^i B ) ) = (/) )
3	1 2	eqtr3id	|- ( ( A i^i B ) = (/) -> ( ( R |` A ) i^i ( R |` B ) ) = (/) )