Definition:Existential Quantifier/Unique/Definition 2

Definition
There exists a unique object $x$ such that $\map P x$, denoted $\exists ! x: \map P x$, :
 * $\exists x : \forall y : \paren {\map P y \iff x = y}$

Also see

 * Equivalence of Definitions of Unique Existential Quantifier