What is the definition of an $n$-ary relation "not depending upon" its $m$-th coordinate?

Question

The function $f$ from $\mathbb{R}^2$ to $\mathbb{R}$ defined by the equation $f(x,y)=x+5$ does not depend upon the 2nd coordinate. I know how to define it, like so: For all $x,y_1,y_2, \in \mathbb{R}, f(x,y_1)=f(x,y_2)$. And given a set $S$ and positive integers $n$ and $m$ with $m \leq n$, it is similarly easy to define when a function $f$ from a cartesian power $S^n$ to $S$ does not depend upon its $m$-th coordinate. However, I don't know how to formally define when a relation $R$ on $S^n$ does not depend upon its $m$-th coordinate. For example, consider the ternary relation $R$ defined on $\mathbb{R}^3$ by the condition $R(x,y,z) \leftrightarrow x+z > 0$. This does not depend upon its 2nd coordinate. But what is the correct definition of this notion?

Answer 1

For a ternary relation $R(x, y, z)$, independence of the $2^{\mathrm{nd}}$ coordinate is given formally by:

$$ \forall x, y_1, y_2, z(R(x, y_1, z) \Leftrightarrow R(x, y_2, z)) $$

I.e., the truth value of the relation is the same for two triples that differ only in the $2^{\mathrm{nd}}$ coordinate.