Definition:Mapping/Definition 2

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:
 * $\forall x \in S: \forall y_1, y_2 \in T: \left({x, y_1}\right) \in f \land \left({x, y_2}\right) \in f \implies y_1 = y_2$

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

Also see

 * Equivalence of Definitions of Mapping