Image of Intersection under Mapping

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

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

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

Then:
 * $f \left[{S_1 \cap S_2}\right] \subseteq f \left[{S_1}\right] \cap f \left[{S_2}\right]$

That is, the image of the intersection of subsets of a mapping is a subset of the intersection of their images.

Proof
As $f$, being a mapping, is also a relation, we can apply Image of Intersection under Relation:


 * $\mathcal R \left[{S_1 \cap S_2}\right] \subseteq \mathcal R \left[{S_1}\right] \cap \mathcal R \left[{S_2}\right]$

Also see

 * Image of Intersection under One-to-Many Relation
 * Image of Intersection under Injection


 * Preimage of Intersection under Mapping


 * Image of Union under Mapping
 * Preimage of Union under Mapping