Injection of Finite Set is Bijection

Theorem

Let $S$ and $T$ be finite sets such that there exists a bijection $\phi:S\to T$.

Then any injection $f:S\to T$ is a bijection.

Proof

Take any injection $f:S\to T$, then $|S|\leq|f(S)|\leq|T|$, but since $S$ y $T$ are bijective, $|f(S)|=|T|$ and thus $f(S)=T$ because $f(S)\subseteq T$ are finite sets. $\blacksquare$