Definition:Mapping/Definition 3

Definition
Let $S$ and $T$ be sets. A mapping $f$ from $S$ to $T$, denoted $f: S \to T$, is a relation $f \subseteq S \times T$ such that:
 * $\forall \left({x_1, y_1}\right), \left({x_2, y_2}\right) \in f: y_1 \ne y_2 \implies x_1 \ne x_2$

and
 * $\forall x \in S: \exists y \in T: \left({x, y}\right) \in f$

Also see

 * Equivalence of Definitions of Mapping