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 $T$.

Let:
 * $T \sim \left|{T}\right|$

Then:
 * $\left|{S}\right| \le \left|{T}\right|$

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