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Metamath Proof Explorer

Theorem petinidres

Description: A class is a partition by an intersection with the identity class restricted to it if and only if the cosets by the intersection are in equivalence relation on it. Cf. br1cossinidres , disjALTVinidres and eqvrel1cossinidres . (Contributed by Peter Mazsa, 31-Dec-2021)

Ref Expression
Assertion petinidres ( ( 𝑅 ∩ ( I ↾ 𝐴 ) ) Part 𝐴 ↔ ≀ ( 𝑅 ∩ ( I ↾ 𝐴 ) ) ErALTV 𝐴 )

Proof

Step Hyp Ref Expression
1 petinidres2 ⊢ ( ( 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 )