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.