Image of Intersection under Injection

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

Let $$A$$ and $$B$$ be subsets of $$S$$. Then:


 * $$f \left({A \cap B}\right) = f \left({A}\right) \cap f \left({B}\right)$$

iff $$f$$ is an injection.

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

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

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

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 One-to-Many Image of Intersections applies:


 * $$\mathcal{R} \left({A \cap B}\right) = \mathcal{R} \left({A}\right) \cap \mathcal{R} \left({B}\right)$$

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

Given that $$f$$ is a mapping, it follows that:
 * $$f \left({A \cap B}\right) = f \left({A}\right) \cap f \left({B}\right)$$

iff $$f$$ is an injection.

Proof of Generalized Result
Follows directly from the same approach as the above, and from One-to-Many Image of Intersections: Generalized Result.