Pairs in the Graph of a Point of $\mathbb{R}^{n}$

In the setting of Euclidean Space and Real Matrices: Standing Notation, let $n\in\mathbb{N}$, let $x\in\mathbb{R}^{n}$ be a point of Euclidean space, and let $k\in[n]$.

(i) For every set $v$, the ordered pair $(k,v)$ is an element of $x$ if and only if $v$ is the $k$-th component $x_{k}$ of $x$:

$$(k,v)\in x\iff v=x_{k}.$$