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:


 * $\forall A, B \subseteq S: \RR \sqbrk {A \cap B} = \RR \sqbrk A \cap \RR \sqbrk B$

$\RR$ 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:
 * $\forall A, B \subseteq S: f \sqbrk {A \cap B} = f \sqbrk A \cap f \sqbrk B$

$f$ is an injection.