Cardinality of Image of Set not greater than Cardinality of Set

Theorem
Let $X, Y$ be sets.

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

Let $A$ be a subset of $X$.

Then $\left\vert{f^\to\left({A}\right)}\right\vert \le \left\vert{A}\right\vert$

where $\left\vert{A}\right\vert$ denotes the cardinality of $A$.

Proof
By definitions of surjection and restriction of mapping:
 * $F \restriction_A: A \to f^\to\left({A}\right)$ is a surjection

Thus by Surjection iff Cardinal Inequality:
 * $\left\vert{f^\to\left({A}\right)}\right\vert \le \left\vert{A}\right\vert$