Characteristic Function of Universe

Theorem
Let $S$ be a set.

Let $\chi_S: S \to \left\{{0, 1}\right\}$ be its characteristic function (in itself).

Then $\chi_S = f_1$, where $f_1: S \to \left\{{0, 1}\right\}$ is the constant mapping with value $1$.

Proof
From Characteristic Function Determined by 1-Fiber, $\chi_S$ is the mapping determined by:


 * $\forall s \in S: \chi_S (s) = 1 \iff s \in S$

which plainly means that:


 * $\forall s \in S: \chi_S (s) = 1$

and by definition of constant mapping, we conclude that:


 * $\chi_S = f_1$.