It is known that: $ ``\forall x \exists H\_x"$, where $H\_x$ is defined as the set of all sets hereditarily subnumerous to $x$, is a theorem of $\text{ZF}$. The formal definition of $H\_x$ is:


$$H\_x=\{y| \forall z \in TC(\{y\}) (\exists f (f:z \to x \wedge f \text{ is injective}))\}$$, where $TC(k)$ stands for the "Transitive closure of($k$)" defined in the standard manner as the minimal transitive superset of $k$. 



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> Question: is $``\forall x \exists H\_x"$ a thorem of "$\text{ZF-Regularity + Aczel's anti-foundation axiom}$"?
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Does a hereditary size set exist for every set in ZF-Regularity+Aczel's anti-foundation axiom?
The answer is yes. 


The reason is that in AFA, every set is determined by the isomorphism type of the $\in$ relation on its transitive closure. So if a set $y$ hereditarily injects into another set $x$, then the transitive closure of $y$ is determined by a certain graph relation on the nodes coming from the graph of $x$. By suitable applications of the power set axiom and separation, the collection of all such graphs exists as a set, and the sets to which they correspond exists by the replacement axiom. So $H\_x$ exists. 


