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 $f \sqbrk X$ is a finite set.

Proof
By definition of surjection:
 * $f: X \to f \sqbrk X$ is a surjection.

The case when $X \ne \O$:

By Surjection iff Cardinal Inequality:
 * $\card {f \sqbrk X} \le \card X$

Thus by Set of Cardinality not Greater than Cardinality of Finite Set is Finite:
 * $f \sqbrk X$ is finite.

The case when $X = \O$:

By the corollary to Image of Empty Set is Empty Set:
 * $f \sqbrk X = \O$

Thus the result holds.