Images and Preimages under a Bijection and Its Inverse

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $X$ and $Y$ be sets, let $f:X\to Y$ be a bijection, let $g=f^{-1}:Y\to X$ be its inverse, and let $E\subseteq X$. Images and preimages are as in The Image and the Preimage of a Class under a Class (i) and The Image and the Preimage of a Class under a Class (ii).

(i) The image under $f$ is the preimage under $g$.

$$f[E]=g^{-1}[E].$$

(ii) The preimage of the image is the set itself.

$$f^{-1}\bigl[f[E]\bigr]=E.$$