Set is Countable if Cardinality equals Cardinality of Countable Set

Theorem
Let $X, Y$ be sets.

Let:
 * $\left\vert{X}\right\vert = \left\vert{Y}\right\vert$

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

If $X$ is countable then $Y$ is countable.

Proof
Assume that $X$ is countable.

By definition of countable set there exists an injection:L
 * $f: X \to \N$

By definition of cardinality the sets $Y$ and $X$ are equivalent:
 * $Y \sim X$

Then by definition of set equivalence there exists a bijection:
 * $g: Y \to X$

By definition of bijection:
 * $g$ is an injection.

Hence by Composite of Injections is Injection:
 * $f \circ g: Y \to \N$ is an injection.

Thus by definition:
 * $Y$ is countable.