Image of Set Difference under Mapping

Theorem
The image of the set difference is a subset of the set difference of the images.

That is:

Let $$f: S \to T$$ be a mapping. Let $$S_1$$ and $$S_2$$ be subsets of $$S$$.

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

where $$\setminus$$ denotes set difference.

Proof
As $$f$$, being a mapping, is also a relation, we can apply Image of Set Difference:


 * $$\mathcal{R} \left({S_1 \setminus S_2}\right) \subseteq \mathcal{R} \left({S_1}\right) \setminus \mathcal{R} \left({S_2}\right)$$