Graph of Real Bijection in Coordinate Plane intersects Horizontal Line at One Point

Theorem
Let $f: \R \to \R$ be a real function which is bijective.

Let its graph be embedded in the Cartesian plane $\mathcal C$:


 * Graph-of-bijection-intersect-horizontal.png

Every horizontal line through a point $b$ in the codomain of $f$ intersects the graph of $f$ on exactly one point $P = \tuple {a, b}$ where $b = \map f a$.

Proof
By definition, a bijection is a mapping which is both an injection and a surjection.

Let $\mathcal L$ be a horizontal line through a point $b$ in the codomain of $f$.

From Graph of Real Surjection in Coordinate Plane intersects Every Horizontal Line:
 * $\mathcal L$ intersects the graph of $f$ on at least one point $P = \tuple {a, b}$ where $b = \map f a$.

From Graph of Real Injection in Coordinate Plane intersects Horizontal Line at most Once:
 * $\mathcal L$ intersects the graph of $f$ on at most one point $P = \tuple {a, b}$ where $b = \map f a$.

The result follows.