Reals are Isomorphic to Dedekind Cuts

Theorem
Let $\mathscr D$ be set of all Dedekind cuts of the total order $\left({\Q, \leq}\right)$.

Define a mapping $f: \R \to \mathscr D$
 * $\forall x \in \R: f \left({x}\right) = \left\{{y \in \Q: y < x}\right\}$

Then $f$ is a bijection.

Proof
First, we will prove that
 * $\forall x \in \R: f \left({x}\right) \in \mathscr D$

Let $x \in \R$.

It should be proved that $f \left({x}\right)$ is proper subset of $\Q$ such that
 * $(1): \quad \forall z \in f \left({x}\right): \forall y \in \Q: y \prec z \implies y \in f \left({x}\right)$
 * $(2): \quad \forall z \in f \left({x}\right): \exists y \in f \left({x}\right): x \prec y$

$x \notin f \left({x}\right)$ implies $f \left({x}\right)$ is proper subset of $\Q$.