Relativization of Pairing In Chapter 12 of Jech's Set Theory, he proves that ZF is consistent with ZF minus Regularity by showing that if $V=\bigcup_{\alpha\in Ord} V_\alpha$, then in ZF minus Regularity $\sigma^V$ holds for every axiom $\sigma$ of ZF and hence $V$ is a (transitive) model of ZF. In proving that the relativization of Pairing holds, he writes the following:

Given $a,v\in V$, let $c=\{a,b\}$. The set $c$ is in $V$ and since "$c=\{a,b\}$" is $\Delta_0$, the Pairing Axiom holds in $V$.

I'm certainly misunderstanding something here, but I don't see why the assertion "$c=\{a,b\}$ is $\Delta_0$" is important here. If I understand correctly, the relativization of Pairing to $V$ is
$$\forall a\in V \, \forall b\in V\, \exists c\in V \, \forall x\in V \,(x\in c \leftrightarrow(x=a\lor x=b)).$$
Why does this not just directly follow from $c=\{a,b\}\in V$?
 A: Any statement which is $\Delta_0$ is absolute for a transitive class. That is, for any transitive class $M$ and any $\Delta_0$ statement $\phi(x_1, \ldots, x_n)$, we have $\forall x_1 \in M \ldots \forall x_n \in M (\phi(x_1, \ldots, x_n)^M \iff \phi(x_1, \ldots, x_n))$. This can be proved by induction on $\phi(x_1, \ldots, x_n)$.
We can express $c = \{a, b\}$ as a $\Delta_0$ statement $P(a, b, c) = a \in c \land b \in c \land \forall d \in c (d = a \lor d = b)$.
Now consider some statement $Q(x_1, \ldots x_n, y)$ such that we can demonstrate $\forall x_1 \ldots \forall x_n \exists! y Q(x_1, \ldots, x_n, y)$. Then we can introduce new notation $f(x_1, \ldots, x_n)$ which represents the unique $y$ in question.
Suppose further that $Q(x_1, \ldots, x_n, y)$ is absolute for some class $M$.
Then the statement $\Phi := (\forall x_1 \ldots \forall x_n \exists! y (Q(x_1, \ldots, x_n, y)))^M$ is equivalent to stating $\forall x_1 \in M \ldots \forall x_n \in M \exists! y \in M (Q(x_1, \ldots, x_n, y))$. Now the only $y$ such that $Q(x_1, \ldots, x_n, y)$ is $y = f(x_1, \ldots, x_n)$. Therefore, $\Phi$ is exactly the statement that $M$ is closed under $f$.
Putting these two together, we see that $P(a, b, c)$ is $\Delta_0$ and thus is absolute for a transitive class. Therefore, the axiom of pairing holds for a transitive class $M$ if and only if $M$ is closed under the map $a, b \mapsto \{a, b\}$.
In this case, we know that $V$ is a transitive class which is closed under $a, b \mapsto \{a, b\}$. Therefore, $V$ satisfies the axiom of pairing.
Other axioms which fall into this scheme include the axiom of union and the axiom scheme of $\Delta_0$ separation. However, the axiom of power set notably does not fall into this scheme - a transitive class can satisfy the axiom of power sets without actually being closed under power sets. An example of this is $L$, which, assuming $V \neq L$, is not closed under the true power set operation but nevertheless satisfies the axiom of power sets.
