Mapping is Constant iff Image is Singleton

Theorem
A mapping is a constant mapping iff its image is a singleton.

Proof

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

$$\forall x \in S: f_c \left({x}\right) = c \Longrightarrow \mathrm{Im} \left({S}\right) = \left\{{c}\right\}$$

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


 * Now, let $$f: S \to T$$ such that its image is a singleton $$\left\{{c}\right\} \subseteq T$$.

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