This question was posted on Math Stack Exchange, but did not attract an answer. Here is the question:

# Informal Description

Let me start with an example. Let $X$ be the set $\{a, b, c, d, e\}$ and $E$ be the set $\{a, b, c\}$. Let $f$ be a function with domain $X$. Then the mapping that sends $E$ to $\{f(a), f(b), f(c) \}$ is called the direct image of $E$ under $f$, denoted by $f_*(E)$.

Now, let $W$ be the set $\{\{a, b\}, \{c\}\}$. What do you call the mapping that sends $W$ to $\{\{f(a), f(b)\}, \{f(c)\}\}$ ?

Informally, this mapping replaces every element of $X$ *"inside"* $W$ with its image under $f$.

# Slightly More Formal Description

Let us assume the ZFA-axiomatisation of Set Theory, a variant of ZF that allows non-set objects called atoms. Let $A$ be a set of atoms.

Define $P^0(A)$ = A.

And for all $n \in \mathbb{N} \colon$ define $P^n(A)= \mathscr{P}(P^{n-1}(A)) \cup A $ ,

where $\mathscr{P}(-)$ denotes the powerset of a set.

And define $P^{\infty}(A)$ as $\bigcup_{n \in \mathbb{N}} P^{n}(A)$

Let $X$ be a set of atoms. Let $f$ be a function with domain $X$. We could then inductively define a function $f_{**}$ such that $dom(f_{**}) = P^{\infty}(X) \backslash X$ , and $f_{**}$ has the property that for all $Z \in dom(f_{**}) \colon f_{**}(Z) = \{f_{**}(T) | T \in Z \backslash X \} \cup \{f(x) | x \in Z \cap X \}$.

## Examples

Let $X = \{a, b, c, d, e\}$ be a set of atoms. Then $\{a, \{b\}\} \in P^{\infty}(X) \backslash X$ and $f_{**}(\{a, \{b\}\}) = \{f(a), \{f(b)\}\}$

Let $X = \{a, b, c, d, e\}$ be a set of atoms. Then $\{\{ \{c \} \}\} \in P^{\infty}(X) \backslash X$ and $f_{**}(\{\{\{c \} \}\}) = \{\{\{f(c) \} \}\}$

So $f_{**}$ can, in a sense, be thought of as a generalisation of the direct image. My quesiton is, what do you call this concept? And can you point me towards any resources regarding it? I would prefer if the resource makes use of the ZFA-axiomatisation, but I would also accept any other resource.

## Sidenote

I am aware that this concept comes up in Permutation Models for the case when $f$ is a bijective endofunction.

inducesa function on $P^\infty(A)$. $\endgroup$