Theorem. Given a subset $G$ of the Cartesian product $X \times Y$, where $X, Y$ are sets, and $G$ has the property that for each $x \in X$ the set $\{ y \in Y : (x, y) \in G \}$ is the singleton set. Then there's exactly one function $f:X \rightarrow Y$ whose graph is equal to G.

Proof. Since elements of $G$ obey the vertical line test $(\forall x \in X) \: P(x,y) = (z \in \{ y \in Y : (x,y) \in G \} \leftrightarrow z = y )$ we have that $$P(x,y) \leftrightarrow y = f'(x).$$ If there was another function $g \neq f$ whose graph is equal to G this would contradict the results of the 1st part of this exercise: $G_f = G \leftrightarrow f = f'$.
$\Box$

2015 FEB 19 (v1.0)
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