Description: A set is a binary relation if and only if it belongs to the powerclass of the cartesian square of the universal class. (Contributed by Peter Mazsa, 14-Jun-2018) (Revised by BJ, 16-Dec-2023)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | pwvrel | |- ( A e. V -> ( A e. ~P ( _V X. _V ) <-> Rel A ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elpwg | |- ( A e. V -> ( A e. ~P ( _V X. _V ) <-> A C_ ( _V X. _V ) ) ) |
|
| 2 | df-rel | |- ( Rel A <-> A C_ ( _V X. _V ) ) |
|
| 3 | 1 2 | syl6bbr | |- ( A e. V -> ( A e. ~P ( _V X. _V ) <-> Rel A ) ) |