 Rieger-Bernays permutation model

A definable class bijection $\pi$ of a model's universe changes membership as displayed. Images and inverse images of sets under $\pi$ remain sets by the Axiom schema of replacement. The new extension of $y$ is the old set $\pi (y)$. Thus the representative of any desired extension $b$ is ${\pi }^{-1}(b)$. Extensionality follows from bijectivity; separation and replacement translate to old definable formulas; pairs, unions and power sets have explicit representatives. Infinity follows by iterating the new empty set and successor. Foundation need not survive: interchanging $\varnothing$ and $\{\varnothing \}$ makes the old empty set a Quine atom. A full such permutation of a model of choice still satisfies choice; failure of choice requires an additional symmetry restriction.