Image of Intersection under Mapping

Theorem
Let $$f: S \to T$$ be a mapping. Let $$A$$ and $$B$$ be subsets of $$S$$. Then:

$$f \left({A \cap B}\right) \subseteq f \left({A}\right) \cap f \left({B}\right)$$

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

$$\mathcal{R} \left({A \cap B}\right) \subseteq \mathcal{R} \left({A}\right) \cap \mathcal{R} \left({B}\right)$$