Theorem partsuc2

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

	Ref	Expression
Assertion	partsuc2	⊢ ( ( ( 𝑅 ↾ ( 𝐴 ∪ { 𝐴 } ) ) ∖ ( 𝑅 ↾ { 𝐴 } ) ) Part ( ( 𝐴 ∪ { 𝐴 } ) ∖ { 𝐴 } ) ↔ ( 𝑅 ↾ 𝐴 ) Part 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ressucdifsn2	⊢ ( ( 𝑅 ↾ ( 𝐴 ∪ { 𝐴 } ) ) ∖ ( 𝑅 ↾ { 𝐴 } ) ) = ( 𝑅 ↾ 𝐴 )
2		sucdifsn2	⊢ ( ( 𝐴 ∪ { 𝐴 } ) ∖ { 𝐴 } ) = 𝐴
3		parteq12	⊢ ( ( ( ( 𝑅 ↾ ( 𝐴 ∪ { 𝐴 } ) ) ∖ ( 𝑅 ↾ { 𝐴 } ) ) = ( 𝑅 ↾ 𝐴 ) ∧ ( ( 𝐴 ∪ { 𝐴 } ) ∖ { 𝐴 } ) = 𝐴 ) → ( ( ( 𝑅 ↾ ( 𝐴 ∪ { 𝐴 } ) ) ∖ ( 𝑅 ↾ { 𝐴 } ) ) Part ( ( 𝐴 ∪ { 𝐴 } ) ∖ { 𝐴 } ) ↔ ( 𝑅 ↾ 𝐴 ) Part 𝐴 ) )
4	1 2 3	mp2an	⊢ ( ( ( 𝑅 ↾ ( 𝐴 ∪ { 𝐴 } ) ) ∖ ( 𝑅 ↾ { 𝐴 } ) ) Part ( ( 𝐴 ∪ { 𝐴 } ) ∖ { 𝐴 } ) ↔ ( 𝑅 ↾ 𝐴 ) Part 𝐴 )