Image of Set Difference under Injection

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

Let $S_1$ and $S_2$ be subsets of $S$.

Let $S_1 \setminus S_2$ denote the set difference between $S_1$ and $S_2$.

Then:
 * $\forall S_1, S_2 \subseteq S: f \left({S_1}\right) \setminus f \left({S_2}\right) = f \left({S_1 \setminus S_2}\right)$

iff $f$ is an injection.

Proof
From Mapping Image of Set Difference:
 * $f \left({S_1}\right) \setminus f \left({S_2}\right) \subseteq f \left({S_1 \setminus S_2}\right)$