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Metamath Proof Explorer

Theorem ressucdifsn2

Description: The difference between restrictions to the successor and the singleton of a class is the restriction to the class, see ressucdifsn . (Contributed by Peter Mazsa, 24-Jul-2024)

		Ref	Expression
	Assertion	ressucdifsn2	⊢ ( ( 𝑅 ↾ ( 𝐴 ∪ { 𝐴 } ) ) ∖ ( 𝑅 ↾ { 𝐴 } ) ) = ( 𝑅 ↾ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		disjcsn	⊢ ( 𝐴 ∩ { 𝐴 } ) = ∅
2		disjresundif	⊢ ( ( 𝐴 ∩ { 𝐴 } ) = ∅ → ( ( 𝑅 ↾ ( 𝐴 ∪ { 𝐴 } ) ) ∖ ( 𝑅 ↾ { 𝐴 } ) ) = ( 𝑅 ↾ 𝐴 ) )
3	1 2	ax-mp	⊢ ( ( 𝑅 ↾ ( 𝐴 ∪ { 𝐴 } ) ) ∖ ( 𝑅 ↾ { 𝐴 } ) ) = ( 𝑅 ↾ 𝐴 )