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.