I'm studying Chapter 9, titled The Raisonnier Filter in Ralf Schindler's textbook (Set Theory: Exploring Independence and Truth, 2014).
The goal of this chapter is to prove Shelah's theorem: Assume every Σ13 set of reals is Lebesgue measurable. Then $\omega^V$ is inaccessible to the reals.
The argumentation goes in the following way: We assume $ℵ_1$ is not inaccessible to the reals and all $Σ_{12}$ sets are Lebesgue measurable. Then one shows that there is a rapid filter F which is $Σ_{13}$. One finishes by proving that rapid filters are not Lebesgue measurable.
I have a technical question to one of the claims: Let $a\in {}^\omega \omega$ such that $\omega_1^V = \omega_1^{L[a]}$.
Let $f: \omega \rightarrow \omega$ be monotone.
Define A:= $\{(x,y) \in ({}^\omega \omega)^2 \cap L[a,f]: x < _{L[a,f]} y\}$.
Claim: For each $y\in {}^\omega \omega \cap L[a,f] $, $\{x: (x,y)\in A \}$ is countable.
I don't understand why this set is countable. Can anyone explain this to me?
Thank you very much for your help in advance!