Equality of Mappings

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


 * $(1): \quad S_1 = S_2$
 * $(2): \quad T_1 = T_2$
 * $(3): \quad \forall x \in S_1: \map {f_1} x = \map {f_2} x$

Proof
This follows directly from Equality of Relations.

Also defined as
It is worth labouring the point that for two mappings to be equal, not only must their domains be equal, but so must their codomains:
 * It may seem like nit-picking to distinguish between functions which have different [ codomains but are otherwise equal] (and indeed until recently most authors did not) but failure to make the distinction sometimes leads to confusion.

Thus it is worth noting that this is a modern departure, and many earlier sources, for example, do not impose this condition, stating merely that two mappings are equal if the domains are equal and every element of the domain has the same image under each mapping.

Other sources, for example, gloss over the concepts of domain and codomain and merely state the equality of the images.

The influential goes further, and identifies a mapping with its graph, preferring to dispense with a need to specify either its domain or its codomain.