Definition:Image (Relation Theory)/Relation/Element/Singleton

First Definition
Let $F$ be a relation.

$F\left({x}\right)$ is the unique $y$ such that $x F y$.

To achieve this behavior, $z \in F(x)$ if and only if $z \in y$ for a the unique $y$ satisfying $x F y$.


 * $F\left({x}\right) = \left\{ z : \exists y: \left({ z \in y \land x F y }\right) \land \exists ! y: x F y \right\}$

Historical Note
The first definition was first used by Bertrand Russell.