Theorem petxrnidres

Description: A class is a partition by a range Cartesian product with the identity class restricted to it if and only if the cosets by the range Cartesian product are in equivalence relation on it. Cf. br1cossxrnidres , disjALTVxrnidres and eqvrel1cossxrnidres . (Contributed by Peter Mazsa, 31-Dec-2021)

	Ref	Expression
Assertion	petxrnidres	⊢ ( ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) Part 𝐴 ↔ ≀ ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) ErALTV 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		petxrnidres2	⊢ ( ( Disj ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) ∧ ( dom ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) / ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) ) = 𝐴 ) ↔ ( EqvRel ≀ ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) ∧ ( dom ≀ ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) / ≀ ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) ) = 𝐴 ) )
2		dfpart2	⊢ ( ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) Part 𝐴 ↔ ( Disj ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) ∧ ( dom ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) / ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) ) = 𝐴 ) )
3		dferALTV2	⊢ ( ≀ ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) ErALTV 𝐴 ↔ ( EqvRel ≀ ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) ∧ ( dom ≀ ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) / ≀ ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) ) = 𝐴 ) )
4	1 2 3	3bitr4i	⊢ ( ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) Part 𝐴 ↔ ≀ ( 𝑅 ⋉ ( I ↾ 𝐴 ) ) ErALTV 𝐴 )