Equivalence of Definitions of Injection/Definition 1 iff Definition 3

Proof
Let $f: S \to T$ be an injection by definition 1.

Thus:
 * $\forall x_1, x_2 \in S: \map f {x_1} = \map f {x_2} \implies x_1 = x_2$

First we note that:
 * $t \in \Img f \implies \exists x \in \Dom f: \map f x = t$

thus fulfilling the condition for $f^{-1} {\restriction_{\Img f} }$ to be left-total.

Now let:
 * $t \in \Img f: \tuple {t, y}, \tuple {t, z} \in f^{-1}$

Thus:

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

So $f$ is an injection by definition 3.

Let $f: S \to T$ be an injection by definition 3.

Then:
 * $f^{-1} {\restriction_{\Img f} }: \Img f \to \Dom f$ is a mapping

where $f^{-1} {\restriction_{\Img f} }$ is the restriction of the inverse of $f$ to the image set of $f$.

We need to show that:
 * $\forall x, z \in \Dom f: \map f x = \map f z \implies x = z$

So, pick any $x, z \in \Dom f$ such that:
 * $\map f x = \map f z$

Then:

So $f$ is an injection by definition 1.