Why an internal map, maps an internal set to an internal set?

Question

In this paper page 129, assumed as a fact that an internal map, maps internal subsets to internal subsets but I can't see why this happens.....

Answer 1

Any function $f$ in $M$ trivially has the property that the image of every set under $f$ is a set. We can express the property in the formal language of $M$ (since said language is higher-order, consult page 125, paragraph 3), as e.g. $\forall S. \exists S'. \forall x. x \in S' \leftrightarrow \exists y \in S. f(y) = x$.

By the Extension Principle (Section 2.2. of page 126), every property of $f$ which is expressible in the language of $M$ also holds for its extension in $\!^{\star}M$, with the caveat that the quantifiers now range over internal entities (page 125, bullet point iii). So we obtain the property that the image of every internal set under the standard function $f$ is itself an internal set.

A similar argument (using an additional $\forall f$ and appealing to the permanence principle instead of extension) allows us to conclude that generally the image of every internal set under an internal function is itself internal.