Metamath Proof Explorer


Theorem pwvrel

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 ) )

Proof

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 bitr4di
 |-  ( A e. V -> ( A e. ~P ( _V X. _V ) <-> Rel A ) )