Graph of Real Function in Cartesian Plane intersects Vertical at One Point

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

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


 * Graph-of-function-intersect-vertical.png

Every vertical line through a point $a$ in the domain of $f$ intersects the graph of $f$ at exactly one point $P = \tuple {a, \map f a}$.

Proof
From Equation of Vertical Line, a vertical line in $\CC$ through the point $\tuple {a, 0}$ on the $x$-axis has an equation $x = a$.

A real function is by definition a mapping.

Hence:
 * $\forall a_1, a_2 \in \Dom f: a_1 = a_2 \implies \map f {a_1} = \map f {a_2}$

where $\Dom f$ denotes the domain of $f$.

Thus for each $a \in \Dom f$ there exists exactly one ordered pair $\tuple {a, y}$ such that $y = \map f a$.

That is, there is exactly one point on $x = a$ which is also on the graph of $f$.

The result follows.