Equivalence of Definitions of Injection/Definition 1 iff Definition 3

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

Proof

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

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

First we note that:

$$t \in \mathrm{Im} \left({f}\right) \Longrightarrow \exists x \in \mathrm{Dom} \left({f}\right): f \left({x}\right) = t$$.

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

Now:

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


 * Now, let $$f^{-1}$$ be a mapping.

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