Theorem disjresundif

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

	Ref	Expression
Assertion	disjresundif	|- ( ( A i^i B ) = (/) -> ( ( R |` ( A u. B ) ) \ ( R |` B ) ) = ( R |` A ) )

Proof

Step	Hyp	Ref	Expression
1		resundi	|- ( R |` ( A u. B ) ) = ( ( R |` A ) u. ( R |` B ) )
2	1	difeq1i	|- ( ( R |` ( A u. B ) ) \ ( R |` B ) ) = ( ( ( R |` A ) u. ( R |` B ) ) \ ( R |` B ) )
3		difun2	|- ( ( ( R |` A ) u. ( R |` B ) ) \ ( R |` B ) ) = ( ( R |` A ) \ ( R |` B ) )
4	2 3	eqtri	|- ( ( R |` ( A u. B ) ) \ ( R |` B ) ) = ( ( R |` A ) \ ( R |` B ) )
5		disjresdif	|- ( ( A i^i B ) = (/) -> ( ( R |` A ) \ ( R |` B ) ) = ( R |` A ) )
6	4 5	eqtrid	|- ( ( A i^i B ) = (/) -> ( ( R |` ( A u. B ) ) \ ( R |` B ) ) = ( R |` A ) )