If a transitive proper class is a model of ZFC, is it also a model when constructed "inside" another model? – math.stackexchange.com

Let $C = \{x : \phi(x)\}$ be a transitive class model of ZFC, and $M$ be a (set) transitive model of ZFC, where the $\in$ relation in both $C$ and $M$ are the "real" $\in$ relation. Is $C$ ...