Preimages of Unions, Intersections and Complements

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $S$ and $T$ be sets and let $u:S\to T$ be a map; for a subset $V$ of $T$, $u^{-1}(V)$ is its preimage, a subset of $S$, as in Sets and Maps: Ordinary Notation (vii). Let $\mathcal{P}(S)$ and $\mathcal{P}(T)$ be the power sets of $S$ and $T$, and let $g:\mathcal{P}(T)\to\mathcal{P}(S)$, $V\mapsto u^{-1}(V)$, be the map of Preimages under a Map: Membership, the Preimage Map, Sequences of Preimages, and the Set of Sets with Preimage in a Given Set (iii). Let $I$ be a set and $V:I\to\mathcal{P}(T)$, $i\mapsto V_{i}$, a map. Unions and intersections over $I$ are those of the families $(V_{i})_{i\in I}$ and $(u^{-1}(V_{i}))_{i\in I}$, the latter being the composite $g\circ V:I\to\mathcal{P}(S)$, as in Indexed Families of Sets and Their Union, Intersection and Product (ii) and Indexed Families of Sets and Their Union, Intersection and Product (iii); differences of sets are those of The Boolean Operations on Classes, Disjointness, and the Universal Class (i). A sequence $(V_{k})$ in $\mathcal{P}(T)$ is the case $I=\mathbb{N}$, where $I\neq\emptyset$ as $1\in\mathbb{N}$ by Natural Numbers Are the Successors in Omega: One Is Least and Not a Successor of a Natural Number, the Successor Is Injective, and N Is Closed under Addition and Multiplication (ii).

(i) $\bigcup_{i\in I}V_{i}$ is a subset of $T$, and $u^{-1}\bigl(\bigcup_{i\in I}V_{i}\bigr)=\bigcup_{i\in I}u^{-1}(V_{i})$.

(ii) If $I\neq\emptyset$, then $\bigcap_{i\in I}V_{i}$ is a subset of $T$, and $u^{-1}\bigl(\bigcap_{i\in I}V_{i}\bigr)=\bigcap_{i\in I}u^{-1}(V_{i})$.

(iii) For every subset $V$ of $T$, $u^{-1}(T\setminus V)=S\setminus u^{-1}(V)$.

(iv) $u^{-1}(T)=S$.