Description: Partition implies equivalence relation by the cosets of the relation on its natural domain, cf. partim2 . (Contributed by Peter Mazsa, 17-Sep-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | partim | |- ( R Part A -> ,~ R ErALTV A ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | partim2 | |- ( ( Disj R /\ ( dom R /. R ) = A ) -> ( EqvRel ,~ R /\ ( dom ,~ R /. ,~ R ) = A ) ) |
|
| 2 | dfpart2 | |- ( R Part A <-> ( Disj R /\ ( dom R /. R ) = A ) ) |
|
| 3 | dferALTV2 | |- ( ,~ R ErALTV A <-> ( EqvRel ,~ R /\ ( dom ,~ R /. ,~ R ) = A ) ) |
|
| 4 | 1 2 3 | 3imtr4i | |- ( R Part A -> ,~ R ErALTV A ) |