Set which is Equivalent to Countable Set is Countable

Lemma
Let $S$ be countable set.

Let $T$ be a set.

Let $T$ be in one-to-one correspondence with $S$.

Then $T$ is countable.

Proof
$S$ is countable.

Therefore, $S$ is in one-to-one correspondence with a subset of the natural numbers by a definition of countable set.

$T$ is in one-to-one correspondence with $S$.

Therefore, $T$ is in one-to-one correspondence with a subset of the natural numbers by Composite of Bijections is Bijection.

Accordingly, $T$ is countable by a definition of countable.