Injection implies Cardinal Inequality

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

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

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

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

where $\sim$ denotes set equivalence

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

Proof
Let $f \left[{S}\right]$ denote the image of $S$ under $f$.