Theorem. Two functions $f:X \rightarrow Y, \tilde{f}: X \rightarrow Y$ are equal iff they have the same graph.

Proof. The following statements are equivalent

• $f, \tilde{f}$ are equal.
• For all elements in the common domain of $f, \tilde{f}$ their respective elements in the range are the same.
• The graphs $G(f)$ and $G(\tilde{f})$ are equal.

The equivalence of the last two items is proven in the following.

From the definition of a graph we have that $$ G(f) := \{ (x, f(x)) : x \in X \}. $$ Graphs are sets since they're subsets of the Cartesian product $X \times Y$, which can be shown to be a set according to exercise 3.5.1 and remark 3.5.13. Two sets are equal iff every element of one set is element of the other and vice versa. Given two equal graphs $G(f), G(\tilde{f})$ this requires $$ (\forall a) \: a \in G(f) \leftrightarrow a \in G(\tilde{f}). $$ Since the two functions are defined on the same domain this is equivalent to saying $$ (\forall x) \: (x,f(x)) = (x,\tilde{f}(x)) \leftrightarrow (\forall x) \: f(x) = \tilde{f}(x),$$ where the rhs shows the equivalence of equal graphs and equal functions. $\Box$

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