Equality of Mappings

Theorem
Two mappings $$f_1: S_1 \to T_1, f_2: S_2 \to T_2$$ are equal iff:


 * $$S_1 = S_2$$
 * $$T_1 = T_2$$
 * $$\forall x \in S_1: f_1 \left({x}\right) = f_2 \left({x}\right)$$.

It is worth labouring the point that for two mappings to be equal, not only must their domains be equal, but so must their ranges.

Proof
This follows directly from Equality of Relations.