Image of Countable Set under Mapping is Countable

Definition

Let $S$ be a countable set.

Let $T$ be a set.

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

Then the image of $f$ is countable.

Proof

Let $A$ be the image of $f$.

Let $g: S \to A$ be the restriction of $f$ to the cartesian product $S \times A$.

By Surjection by Restriction of Codomain, $g$ is a surjection.

By Surjection from Natural Numbers iff Countable, there exists a surjection $\phi: \N \to S$.

Since the composition of surjections is a surjection, $g \circ \phi: \N \to A$ is a surjection.

Hence, by Surjection from Natural Numbers iff Countable, it follows that $A$ is countable.

$\blacksquare$