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 ∪ { 𝐴 } )
2	1	reseq2i	⊢ ( 𝑅 ↾ suc 𝐴 ) = ( 𝑅 ↾ ( 𝐴 ∪ { 𝐴 } ) )
3	2	difeq1i	⊢ ( ( 𝑅 ↾ suc 𝐴 ) ∖ ( 𝑅 ↾ { 𝐴 } ) ) = ( ( 𝑅 ↾ ( 𝐴 ∪ { 𝐴 } ) ) ∖ ( 𝑅 ↾ { 𝐴 } ) )
4		ressucdifsn2	⊢ ( ( 𝑅 ↾ ( 𝐴 ∪ { 𝐴 } ) ) ∖ ( 𝑅 ↾ { 𝐴 } ) ) = ( 𝑅 ↾ 𝐴 )
5	3 4	eqtri	⊢ ( ( 𝑅 ↾ suc 𝐴 ) ∖ ( 𝑅 ↾ { 𝐴 } ) ) = ( 𝑅 ↾ 𝐴 )