Rectangles, Sections and the Swap: Set Identities

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $X$ and $Y$ be sets, with the Cartesian products $X\times Y$ and $Y\times X$; for $E\subseteq X\times Y$, $x\in X$ and $y\in Y$ let $E_{x}\subseteq Y$ and $E^{y}\subseteq X$ be the sections of $E$, and likewise for subsets of $Y\times X$. For a function $u:S\to T$ and $V\subseteq T$, $u^{-1}(V)$ is the preimage of $V$, as in Preimages under a Map: Membership, the Preimage Map, Sequences of Preimages, and the Set of Sets with Preimage in a Given Set. Let $A,A'\subseteq X$, $B,B'\subseteq Y$, $E,E'\subseteq X\times Y$, let $(E_{k})_{k\in\mathbb{N}}$ be a sequence of subsets of $X\times Y$, and let $x\in X$ and $y\in Y$. Let $i_{x}:Y\to X\times Y$ and $i^{y}:X\to X\times Y$ be the insertion functions and $t:Y\times X\to X\times Y$ the swap,

$$i_{x}(y')=(x,y'),\qquad i^{y}(x')=(x',y),\qquad t(y',x')=(x',y')\qquad(x'\in X,\ y'\in Y).$$

(i) Rectangles.

$$(A\times B)\cap(A'\times B')=(A\cap A')\times(B\cap B'),$$ $$(A\times B)_{x}=\begin{cases}B,&x\in A,\\\emptyset,&x\notin A,\end{cases}\qquad(A\times B)^{y}=\begin{cases}A,&y\in B,\\\emptyset,&y\notin B.\end{cases}$$

(ii) Sections of complements, unions and intersections.

$$\bigl((X\times Y)\setminus E\bigr)_{x}=Y\setminus E_{x},\qquad\Bigl(\bigcup_{k\in\mathbb{N}}E_{k}\Bigr)_{x}=\bigcup_{k\in\mathbb{N}}(E_{k})_{x},\qquad(E\cap E')_{x}=E_{x}\cap E'_{x};$$

in particular, if $E\cap E'=\emptyset$ then $E_{x}\cap E'_{x}=\emptyset$. The same holds for sections at $y$, with $X\setminus E^{y}$ in place of $Y\setminus E_{x}$.

(iii) Sections are preimages under the insertions.

$$i_{x}^{-1}(E)=E_{x},\qquad(i^{y})^{-1}(E)=E^{y}.$$

(iv) The swap.

$$t^{-1}(A\times B)=B\times A,\qquad\bigl(t^{-1}(E)\bigr)_{y}=E^{y},\qquad\bigl(t^{-1}(E)\bigr)^{x}=E_{x}.$$