Image of Intersection under Injection/Proof 1

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 applies:


 * $\forall A, B \subseteq S: \mathcal R \left[{A \cap B}\right] = \mathcal R \left[{A}\right] \cap \mathcal R \left[{B}\right]$

$\mathcal R$ 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 $f$ is also an injection.

It follows that:
 * $\forall A, B \subseteq S: f \left[{A \cap B}\right] = f \left[{A}\right] \cap f \left[{B}\right]$

$f$ is an injection.