I've read the proof of the independence of the axiom of dependent choice $\text{DC}$ from $\text{ZF}+\text{AC}_\omega$ in Jech's The Axiom of Choice, but there he proves it in $\text{ZFA}$ and then invokes some transfer theorem, making hard to pinpoint the set where $\text{DC}$ fails in the symmetric extension.
My question is how to define a symmetric model in which $\text{ZF}+\text{AC}_\omega+\neg\text{DC}(\mathbb{R})$ holds.
Can I re-use in some "cheap" way the construction defined in Jech's proof I mentioned above?
Any hint?
Thanks!