Image of Mapping from Finite Set is Finite

Theorem
Let $X, Y$ be sets.

Let $f:X \to Y$ be a mapping.

Let $X$ be a finite set.

Then $f\left[{X}\right]$ is a finite set.

Proof
By definition of surjection:
 * $f:X \to f\left[{X}\right]$ is a surjection.

The case when $X \ne \varnothing$:

By Surjection iff Cardinal Inequality:
 * $\left\vert{f\left[{X}\right]}\right\vert \le \left\vert{X}\right\vert$

Thus by Set of Cardinality not Greater than Cardinality of Finite Set is Finite:
 * $f\left[{X}\right]$ is finite.

The case when $X = \varnothing$:

By Image of Empty Set is Empty Set/Corollary:
 * $f\left[{X}\right] = \varnothing$

Thus the resukt holds.