Image of Set Difference under Mapping

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

The image of the set difference of two subsets of $S$ is a subset of the set difference of the images.

That is:

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

Then:
 * $f \sqbrk {S_1} \setminus f \sqbrk {S_2} \subseteq f \sqbrk {S_1 \setminus S_2}$

where $\setminus$ denotes set difference.

Also see

 * Difference of Images under Mapping not necessarily equal to Image of Difference: equality does not hold in general