Indicator Function of a Subset

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $X$ be a set and let $A\subseteq X$.

(i) The indicator function of $A$ is the map $\mathbf{1}_{A}:X\to\mathbb{R}$ with $\mathbf{1}_{A}(x)=1$ for every $x\in A$ and $\mathbf{1}_{A}(x)=0$ for every $x\in X$ with $x\notin A$. It exists and is unique by Maps Defined by Cases (i), applied to the predicative condition $x\in A$ and the expressions $1$ and $0$, both elements of $\mathbb{R}$.