Context of the problem I think it is important to say that this is how predicate quantification is defined in my textbook:
Let $\omega = \mathbb{N} + \{0\}$, $S_i \subseteq \omega, L_i \subseteq \Sigma^{*}$ for an arbitrary alphabet $\Sigma$. If
$$P : S_0 \times S_1 \times \ldots \times S_n \times L_1 \times \ldots \times L_m $$
is a predicate and $S \subseteq S_0$, then
$$(\forall t \in S)_{t \leq x}P(t, \overrightarrow{x}, \overrightarrow{\alpha})$$
is $1$ when $P(t, \overrightarrow{x}, \overrightarrow{\alpha}) = 1$ for all $t \leq x$. The domain of the quantified proposition is
$$\omega \times S_1 \times \ldots \times S_n \times L_1 \times \ldots \times L_m $$
where the first argument (accounted by $\omega$) is the upper bound $x$. We accept without proof the following two statements, where "p.r." abbreviaties "primitive recursive":
If $P: S_0 \times S_1 \times \ldots \times S_n \times L_1 \times\ldots \times L_m \to \omega$ a predicate $\Sigma$-p.r., and $S \subseteq S_0$ is $\Sigma$-p.r., then both quantifications over $P$ are $\Sigma$-p.r.
If $P: S_1 \times \ldots \times S_n \times L_1 \times \ldots \times L_m L_{m+1} \to \omega$ a predicate $\Sigma$-p.r., and $L \subseteq L_{m+1}$ is $\Sigma$-p.r., then both quantifications over $P$ are $\Sigma$-p.r.
I was given the following problem.
Let $\Sigma = \{@, !\}$. Show that $\{(2^x, @^x, !) : x \in \omega \land x \text{ is odd}\}$ is $\Sigma$-p.r. (this is, prove that it has a primitive recursive characteristic function).
Let $S$ denote the set in question. For clarity, observe that a few elements of $S$ are
\begin{align*} (2, @, !), (8, @@@, !), (32, @@@@@, !), \ldots \end{align*}
The set is $\Sigma$-p.r. if and only if there is a primitive recursive function that determines whether a given tuple is in the set. It seems that, to determine if a tuple in $\omega \times \Sigma^{*} \times \Sigma^{*}$ is in $S$, we should use the function
\begin{align*} \mathcal{F} = \lambda xy \alpha \beta \left[ (\exists k \in \omega)_{k \leq x} \left( y = 2^{k+1} \land \alpha = @^{2k+1}\right) \land \beta = ! \right] \end{align*}
However, at least in my textbook, the conjunction of two predicates $P_1 \land P_2$ is defined only when the predicates have identical domains. However, the domain of $\lambda xy [x = 2^{y+ 1}]$ is $\omega^2$ and that of $\lambda x\alpha \left[ \alpha = @^{x+1} \right] $ is $\omega \times \Sigma^{*}$. How could one define a valid, primitive characteristic function of $S$?