Equivalence of Definitions of Injection/Definition 1 iff Definition 3

Theorem
A mapping $f$ is an injection iff:
 * $f^{-1}: \operatorname{Im} \left({f}\right) \to \operatorname{Dom} \left({f}\right)$

is a mapping.

Necessary Condition
Let $f: S \to T$ be an injection.

First we note that:
 * $t \in \operatorname{Im} \left({f}\right) \implies \exists x \in \operatorname{Dom} \left({f}\right): f \left({x}\right) = t$

thus fulfilling the condition $\forall y \in T: \exists x \in S: f \left({x}\right)= y$.

Now let $t \in \operatorname{Im} \left({f}\right): \left({t, y}\right), \left({t, z}\right) \in f^{-1}$.

Thus:

So by the definition of mapping, $f^{-1}$ is a mapping.

Sufficient Condition
Let $f^{-1}$ be a mapping.

We need to show that $\forall x, z \in \operatorname{Dom} \left({f}\right): f \left({x}\right) = f \left({z}\right) \implies x = z$.

So, pick any $x, z \in \operatorname{Dom} \left({f}\right)$ such that $ f \left({x}\right) = f \left({z}\right)$.

Then:

Thus by the definition of an injection, $f$ is an injection.

Comment
Some sources, for example, use this property as the definition of an injection.