Injection is Bijection iff Inverse is Injection

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

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

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.

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.