Inverse of Injective and Surjective Mapping is Mapping/Proof 1

Proof
Recall the definition of the inverse of $f$:

$f^{-1} \subseteq T \times S$ is the relation defined as:
 * $f^{-1} = \set {\tuple {t, s}: t = \map f s}$

Let $f: S \to T$ be a mapping such that:
 * $(1): \quad f$ is an injection
 * $(2): \quad f$ is a surjection.

By Inverse of Injection is Many-to-One Relation, $f^{-1}$ is many-to-one.

From Inverse of Surjection is Relation both Left-Total and Right-Total $\map {f^{-1} } y$ is left-total.

Thus $f^{-1}$ is:
 * many-to-one

and
 * left-total.

Hence, by definition, $f^{-1}$ is a mapping.