In ZF-Regularity, can we define a notion of cardinality such that for all sets $A$ and $B$, $card(A) = card(B)$ iff there exists a bijection from $A$ to $B$?
If we add a function symbol C to the signature of set theory as well as this axiom $\forall A \forall B (C(A)=C(B) \iff \text{there exists a bijection from $A$ to $B$})$ to ZF-Regularity, would this be a conservative extension of ZF-Regularity?
The answers are no and yes. This was proved by Azriel Lévy,
The key point is that allowing atoms (or urelements) is something that can be translated to Quine atoms and the failure of regularity, so it is enough to discuss models with atoms.
Finally, to see that the addition of the operation is conservative, note that given any set $A$, we can take the equipotence relation on $\mathcal P(A)$, which is a bona fide relation, and use it to prove anything we need.