Let $P = (Y, {\preceq})$ be a D1103: Partially ordered set.
Let $f : X \to Y$ be a D18: Map.
A D2218: Set element $m \in f(X)$ is a minimum of $f$ with respect to $P$ if and only if \begin{equation} \forall \, y \in f(X) : m \preceq y \end{equation}

Let $P = (Y, {\preceq})$ be a D1103: Partially ordered set.
Let $f : X \to Y$ be a D18: Map.
A D2218: Set element $m \in f(X)$ is a minimum of $f$ with respect to $P$ if and only if \begin{equation} \forall \, x \in X : m \preceq f(x) \end{equation}