Domain of Injection to Countable Set is Countable

Theorem
Let $X$ be an infinite set.

Let $f: X \to \N$ be an injection.

Then $X$ is countably infinite.

Proof
Let $Y = \left\{{f \left({x}\right): x \in X}\right\}$.

Since $f$ is injective, $Y$ is an infinite subset of $\N$.

Thus from Subset of Countable Set, $Y$ is countably infinite.

The result follows.