Image of Intersection under Mapping

Theorem
The image of the intersection is a subset of the intersection of the images.

That is:

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

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

Generalized Result
Let $$S_i \subseteq S: i \in \N^*_n$$.

Then:
 * $$f \left({\bigcap_{i = 1}^n S_i}\right) \subseteq \bigcap_{i = 1}^n f \left({S_i}\right)$$.

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


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

and


 * $$\mathcal{R} \left({\bigcap_{i = 1}^n S_i}\right) \subseteq \bigcap_{i = 1}^n \mathcal{R} \left({S_i}\right)$$.