Image of Intersection under Injection/Family of Sets

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

Let $\left\langle{S_i}\right\rangle_{i \in I}$ be a family of subsets of $S$.

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

Then:
 * $\displaystyle f \left[{\bigcap_{i \mathop \in I} S_i}\right] = \bigcap_{i \mathop \in I} f \left[{S_i}\right]$

iff $f$ is an injection.

Proof
An injection is a type of one-to-one relation, and therefore also a one-to-many relation.

Therefore Image of Intersection under One-to-Many Relation: Family of Sets applies:


 * $\displaystyle \mathcal R \left[{\bigcap_{i \mathop \in I} S_i}\right] = \bigcap_{i \mathop \in I} \mathcal R \left[{S_i}\right]$

iff $\mathcal R$ is a one-to-many relation.

We have that $f$ is a mapping and therefore a many-to-one relation.

So $f$ is a one-to-many relation iff $f$ is also an injection.

It follows that:
 * $\displaystyle f \left[{\bigcap_{i \mathop \in I} S_i}\right] = \bigcap_{i \mathop \in I} f \left[{S_i}\right]$

iff $f$ is an injection.