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

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

Let $$\mathcal P \left({S}\right)$$ be the power set of $$S$$.

Let $$\mathbb S \subseteq \mathcal P \left({S}\right)$$.

Then:
 * $$f \left({\bigcap \mathbb S}\right) \subseteq \bigcap_{X \in \, \mathbb S} f \left({X}\right)$$

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


 * $$\mathcal R \left({\bigcap \mathbb S}\right) \subseteq \bigcap_{X \in \, \mathbb S} \mathcal R \left({X}\right)$$

Note
Note that equality does not hold in general.

Let:
 * $$S_1 = \left\{{x \in \Z: x \le 0}\right\}$$
 * $$S_2 = \left\{{x \in \Z: x \ge 0}\right\}$$
 * $$f: \Z \to \Z: \forall x \in \Z: f \left({x}\right) = x^2$$

We have:
 * $$f \left({S_1}\right) = \left\{{0, 1, 4, 9, 16, \ldots}\right\} = f \left({S_2}\right)$$

Then:
 * $$f \left({S_1}\right) \cap f \left({S_2}\right) = \left\{{0, 1, 4, 9, 16, \ldots}\right\}$$

but:
 * $$f \left({S_1 \cap S_2}\right) = f \left({\left\{{0}\right\}}\right) = \left\{{0}\right\}$$

Note that from Injection Image of Intersections equality always holds iff $$f$$ is an injection.

Also see One-to-Many Image of Intersections.