Cardinality of Image of Mapping not greater than Cardinality of Domain

Theorem
Let $S$ and $T$ be sets.

Let $f : S \to T$ be a mapping.

Let $\left|{ S }\right|$ denote the cardinal number of $S$.

Let $S \sim \left|{ S }\right|$.

Then:


 * $\left\vert{ \operatorname{Im} \left({f}\right) }\right\vert \le \left\vert{ S }\right\vert$