Equivalence of Definitions of Bijection/Definition 1 iff Definition 2

Proof
From Injection iff Left Inverse, $f$ is an injection $f$ has a left inverse mapping.

From Surjection iff Right Inverse, $f$ is a surjection $f$ has a right inverse mapping.

Putting these together, it follows that:
 * $f$ is both an injection and a surjection


 * $f$ has both a left inverse and a right inverse.
 * $f$ has both a left inverse and a right inverse.