Let $S_N$ be a D1607: Set of permutations on N letters.
Let $X$ be a D11: Set such that

(i)	$X^N : = \prod_{n = 1}^N X$ is a D326: Cartesian product
(ii)	$f : X^N \to Y$ is a D18: Map

Then $f$ is a symmetric map if and only if \begin{equation} \forall \, \pi \in S_N : \forall \, (x_1, \ldots, x_N) \in X^N : f(x_1, \ldots, x_N) = f(x_{\pi(1)}, \ldots, x_{\pi(N)}) \end{equation}