Injection is Bijection iff Inverse is Injection

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

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

Proof

 * 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.


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

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