Mapping is Constant iff Image is Singleton

Theorem

Let $f_c: S \to T$ be a constant mapping. Then from the definition of the image of a element:

$\forall x \in S: \map {f_c} x = c \implies \Img S = \set c$

Thus the image of $f_c: S \to T$ is a singleton.

Let $f: S \to T$ such that its image is a singleton $\set c \subseteq T$.

$\ds \forall x \in S: \, $

$\ds \map f x$

$\in$

$\ds \Img f$

$\ds \leadsto \ \ $

$\ds \forall x \in S: \, $

$\ds \map f x$

$\in$

$\ds \set c$

$\ds \leadsto \ \ $

$\ds \forall x \in S: \, $

$\ds \map f x$

$=$

$\ds c$

Definition of Singleton

Thus $f: S \to T$ is a constant mapping (and we can write it $f_c: S \to T$).

$\blacksquare$