Theorem partsuc2

Description: Property of the partition. (Contributed by Peter Mazsa, 24-Jul-2024)

	Ref	Expression
Assertion	partsuc2	|- ( ( ( R |` ( A u. { A } ) ) \ ( R |` { A } ) ) Part ( ( A u. { A } ) \ { A } ) <-> ( R |` A ) Part A )

Proof

Step	Hyp	Ref	Expression
1		ressucdifsn2	|- ( ( R |` ( A u. { A } ) ) \ ( R |` { A } ) ) = ( R |` A )
2		sucdifsn2	|- ( ( A u. { A } ) \ { A } ) = A
3		parteq12	|- ( ( ( ( R |` ( A u. { A } ) ) \ ( R |` { A } ) ) = ( R |` A ) /\ ( ( A u. { A } ) \ { A } ) = A ) -> ( ( ( R |` ( A u. { A } ) ) \ ( R |` { A } ) ) Part ( ( A u. { A } ) \ { A } ) <-> ( R |` A ) Part A ) )
4	1 2 3	mp2an	|- ( ( ( R |` ( A u. { A } ) ) \ ( R |` { A } ) ) Part ( ( A u. { A } ) \ { A } ) <-> ( R |` A ) Part A )