Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 for relations. (Contributed by Mario Carneiro, 16-Aug-2015) Do not depend on df-cnv , reverse equality. (Revised by BJ, 16-Oct-2026) Note: this theorem could be used in place of opabid2 , which could be deleted.
| Ref | Expression | ||
|---|---|---|---|
| Assertion | bj-dfrel4v | |- ( Rel R <-> { <. x , y >. | x R y } = R ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-dfrel3 | |- ( Rel R <-> ( R |` _V ) = R ) |
|
| 2 | relpart | |- ( R |` _V ) = { <. x , y >. | x R y } |
|
| 3 | 2 | eqeq1i | |- ( ( R |` _V ) = R <-> { <. x , y >. | x R y } = R ) |
| 4 | 1 3 | bitri | |- ( Rel R <-> { <. x , y >. | x R y } = R ) |