Injection is Bijection iff Inverse is Injection

Theorem

Let $\phi: S \to T$ be an injection.

Then $\phi$ is a bijection if and only if its inverse $\phi^{-1}$ is also an injection.

Proof

Necessary Condition

Let $\phi$ be a bijection.

Then from Bijection iff Inverse is Bijection, its inverse $\phi^{-1}$ is also a bijection and therefore by definition an injection.

$\Box$

Sufficient Condition

Let $\phi$ be an injection such that $\phi^{-1}$ is also an injection.

By Cardinality of Surjection, and the Cantor-Bernstein-Schröder Theorem, the result follows.

$\blacksquare$