Characteristic Function of Universe

Theorem

Let $S$ be a set.

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

Then:

$\chi_S = f_1$

where $f_1: S \to \set {0, 1}$ is the constant mapping with value $1$.

Proof

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

$\forall s \in S: \map {\chi_S} s = 1 \iff s \in S$

Thus:

$\forall s \in S: \map {\chi_S} s = 1$

By definition of constant mapping:

$\chi_S = f_1$

$\blacksquare$