I am trying to solve the following problem in ZF: "Show that the collection $\{\kappa \ | \ \kappa=\aleph_\kappa\}$ is a proper class. My thinking is that in the construction any element of this collection, one takes an arbitrary ordinal $\alpha$, and that therefore one might be able to show it is a proper class by showing that there is a bijection between the class of all ordinals and the class given above, since the class of all ordinals is a proper class.
However, I'm not sure how to fully make sense of the concept of a function between proper classes, since a function is normally defined as being between sets. Is this a valid line of reasoning, and if so, how does one formalise this using the axioms of ZF?