Understanding the Proof of Proposition 5.1.3 from Measure Theory by Donald Cohn

Question

I am self-studying Measure Theory by Donald Cohn. I got stuck on a step of his proof of Proposition 5.1.3. Here is the proposition, its proof, and where I got confused:

Proposition 5.1.3$\quad$ Let $(X,\mathscr{A},\mu)$ and $(Y,\mathscr{B},\nu)$ be $\sigma$-finite measure spaces. If $E$ belongs to the $\sigma$-algebra $\mathscr{A}\times\mathscr{B}$, then the function $x\mapsto\nu(E_x)$ is $\mathscr{A}$-measurable and the function $y\mapsto\mu(E^y)$ is $\mathscr{B}$-measurable.

Proof$\quad$ First suppose that the measure $\nu$ is finite. Let $\mathscr{F}$ be the class of those sets $E$ in $\mathscr{A}\times\mathscr{B}$, then the function $x\mapsto\nu(E_x)$ is $\mathscr{A}$-measurable. (Lemma 5.1.2 implies that $E_x$ belongs to $\mathscr{B}$, and hence that $\nu(E_x)$ is defined). If $A\in\mathscr{A}$ and $B\in\mathscr{B}$, then $\nu((A\times B)_x)=\nu(B)\chi_A(x)$, and so the rectangle $A\times B$ belongs to $\mathscr{F}$. In particular, the space $X\times Y$ belongs to $\mathscr{F}$. Note that if $E$ and $F$ are sets in $\mathscr{A}\times\mathscr{B}$ such that $E\subseteq F$, then $\nu((F-E)_x)=\nu(F_x)-\nu(E_x)$, and that if $\{E_n\}$ is an increasing sequence of sets in $\mathscr{A}\times\mathscr{B}$, then $\nu\left(\left(\bigcup_{n=1}^{\infty}E_n\right)_x\right)=\lim_{n\to\infty}\nu((E_n)_x)$; it follows that $\mathscr{F}$ is closed under the formation of proper differences and under the formation of unions of increasing sequences of sets. Thus $\mathscr{F}$ is a $d$-system. Since the family of rectangles with measurable sides is closed under the formation of finite intersections, Theorem 1.6.2 implies that $\mathscr{F}=\mathscr{A}\times\mathscr{B}$. Thus $x\mapsto\nu(E_x)$ is measurable for each $E$ in $\mathscr{A}\times\mathscr{B}$.

I am confused by the last step: In order to apply Theorem 1.6.2 (see below), we need $\mathscr{F}$ to be the $d$-system generated by the $\pi$-system of the family of rectangles with measurable sides. But the author only showed that $\mathscr{F}$ is any $d$-system. Moreover, I am having a difficult time proving that any $d$-system that includes the $\pi$-system of the family of rectangles with measurable sides also includes $\mathscr{F}$.

Could someone please help me out? Thanks a lot in advance!

Definitions and Results

Definition$\quad$ Let $(X,\mathscr{A})$ and $(Y,\mathscr{B})$ be measurable spaces, and as usual, let $X\times Y$ be the Cartesian product of the sets $X$ and $Y$. A subset of $X\times Y$ is a rectangle with measurable sides if it has the form $A\times B$ for some $A$ in $\mathscr{A}$ and some $B$ in $\mathscr{B}$; the $\sigma$-algebra on $X\times Y$ generated by the collection of all rectangles with measurable sides is called the product of the $\sigma$-algebras $\mathscr{A}$ and $\mathscr{B}$ and is denoted by $\mathscr{A}\times\mathscr{B}$.

Definition$\quad$ Suppose that $X$ and $Y$ are sets and that $E$ is a subset of $X\times Y$. Then for each $x$ in $X$ and each $y$ in $Y$ the sections$E_x$ and $E^y$ are the subsets of $Y$ and $X$ given by\begin{align*} E_x = \{y\in Y:(x,y)\in E\}\end{align*}and\begin{align*} E^y = \{x\in X:(x,y)\in E\}.\end{align*}If $f$ is a function on $X\times Y$, then the sections$f_x$ and $f^y$ are the functions on $Y$ and $X$ given by\begin{align*} f_x(y)=f(x,y)\end{align*}and\begin{align*} f^y(x)=f(x,y).\end{align*}

Lemma 5.1.2$\quad$Let $(X,\mathscr{A})$ and $(Y,\mathscr{B})$ be measurable spaces.

1. If $E$ is a subset of $X\times Y$ that belongs to $\mathscr{A}\times\mathscr{B}$, then each section $E_x$ belongs to $\mathscr{B}$ and each section $E^y$ belongs to $\mathscr{A}$.

2. If $f$ is an extended real-valued (or a complex-valued) $\mathscr{A}\times\mathscr{B}$-measurable function on $X\times Y$, then each section $f_x$ is $\mathscr{B}$-measurable and each section $f^y$ is $\mathscr{A}$-measurable.

Definition$\quad$ Let $X$ be a set. A collection $\mathscr{D}$ of subsets of $X$ is a *d-system (or a Dynkin class) on $X$ if

1. $X \in \mathscr{D}$

2. $A-B\in\mathscr{D}$ whenever $A,B\in\mathscr{D}$ and $A\supseteq B$, and

3. $\bigcup_nA_n\in\mathscr{D}$ whenever $\{A_n\}$ is an increasing sequence of sets in $\mathscr{D}$.

Definition$\quad$ A collection of subsets of $X$ is a $\pi$-system on $X$ if it is closed under the formation finite intersection.

Remark$\quad$ Note that the intersection of a nonempty family of $d$-systems on a set $X$ is a $d$-system on $X$ and that an arbitrary collection of subsets of $X$ is included in some $d$-system on $X$, namely the collection of all subsets of $X$. Hence if $\mathscr{C}$ is an arbitrary collection of subsets of $X$, then the intersection of all the $d$-systems on $X$ that include $\mathscr{C}$ is a $d$-system on $X$ that includes $\mathscr{C}$; this intersection of the smallest such $d$-system and is called the $d$-system generated by $\mathscr{C}$. We will sometimes denote this $d$-system by $d(\mathscr{C})$.

Theorem 1.6.2$\quad$Let $X$ be a set, and let $\mathcal{C}$ be a $\pi$-system on $X$. Then the $\sigma$-algebra generated by $\mathcal{C}$ coincides with the $d$-system generated by $\mathcal{C}$.

Updates

What I have been trying is that, let $\mathscr{D}$ be any $d$-system that includes the family $\mathscr{C}$ of rectangles with measurable sides, and then see whether I can prove that if $E\in\mathscr{F}$ then $E\in\mathscr{D}$.

I had a thought, though I don't think it's helpful: I wanted to show that if $E\in\mathscr{A}\times\mathscr{B}$ and if $x\mapsto\nu(E_x)$ is $\mathscr{A}$-measurable, then $E$ is a rectangle with measurable sides. But I kind of already think this might be wrong, for example, considering a triangle on $\mathbb{R}^2$ with the Lebesgue measure.