Theorem ressucdifsn

Description: The difference between restrictions to the successor and the singleton of a class is the restriction to the class. (Contributed by Peter Mazsa, 20-Sep-2024)

	Ref	Expression
Assertion	ressucdifsn	|- ( ( R |` suc A ) \ ( R |` { A } ) ) = ( R |` A )

Proof

Step	Hyp	Ref	Expression
1		df-suc	|- suc A = ( A u. { A } )
2	1	reseq2i	|- ( R |` suc A ) = ( R |` ( A u. { A } ) )
3	2	difeq1i	|- ( ( R |` suc A ) \ ( R |` { A } ) ) = ( ( R |` ( A u. { A } ) ) \ ( R |` { A } ) )
4		ressucdifsn2	|- ( ( R |` ( A u. { A } ) ) \ ( R |` { A } ) ) = ( R |` A )
5	3 4	eqtri	|- ( ( R |` suc A ) \ ( R |` { A } ) ) = ( R |` A )