Injection implies Cardinal Inequality

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

Let $f : S \to T$ be an injection.

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

Suppose $T \sim \left|{ T }\right|$.

Then, $\left|{ S }\right| \le \left|{ T }\right|$

Proof
Let $f " S$ denote the image of $S$ under $f$.