Vopěnka's Principle without Choice: Preservation under Symmetric Extensions

We prove that every set-sized symmetric extension of a model of $\mathsf{ZF}+\mathsf{VP}$ again satisfies $\mathsf{ZF}+\mathsf{VP}$, where $\mathsf{VP}$ is formulated for arbitrary set-sized languages. For each standard $n\geq 1$, the same preservation theorem holds for $\mathsf{VP}(Π_n)$, with set