= Solution
Let $\operatorname{Map}(f,\delta,\kappa)$ be the bounded graph predicate for a function $\delta\to\kappa$. For an infinite <cardinal number>, regularity is expressed by
$$
\boxed{\operatorname{InfCard}(\kappa)\ \land\
\forall f\,\forall\delta\in\kappa\left(
\operatorname{Map}(f,\delta,\kappa)\ \Longrightarrow\
\exists\beta\in\kappa\ \forall\xi\in\delta\ f(\xi)<\beta\right).}
$$
The predicate $\operatorname{InfCard}$ is the cardinal predicate in (i), conjoined with the bounded condition that $\kappa$ is a nonzero <limit ordinal>. For an infinite <cardinal number>, this latter condition holds automatically; it excludes the finite cardinals. The expression $f(\xi)<\beta$ abbreviates bounded graph membership with the output lying in $\beta$.
Every quantifier after $\forall f$ is bounded, and the cardinal predicate is already $\Pi_1$. Their conjunction can be placed in universal form, so this is a <Pi-one formula modulo ZF>. Its mathematical content is that no function from a smaller ordinal is cofinal in $\kappa$. By the definition of <cofinality>, that is equivalent to $\operatorname{cf}(\kappa)=\kappa$. Hence <regular cardinalhood is Pi-one definable>. This uses the standard convention that a <regular cardinal> is infinite.
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