Let $A$ and $B$ groups and $A\wr B$ is a restricted wreath product. Prove that $$ [A\wr B, A\wr B]=[B, B]\ H $$ where $\mathrm{fun}(B, A)=\{f| f:B\longrightarrow A\quad \mathrm{supp}f<+\infty\}$ and $H=\left\{f|f\in\mathrm{fun}(B, A)\ \prod_{b\in B}f(b)\equiv e\pmod{[A,A]}\right\}$ ($A\wr B=B\ \mathrm{fun}(B, A)$ and $bf\cdot b'f'=bb'\ f^{b'}f'$).
This task is from the book Fundamentals of the Theory of Groups.
I did the following: $$ [A\wr B, A\wr B]=[B\ \mathrm{fun}(B, A), B\ \mathrm{fun}(B, A)]= [B, B]\ H $$ and began to consider the binary operation $[xf,\ yg]=[x,\ y]h$ and tried to prove that it was true $\prod_{b\in B}h(b)\equiv e\pmod{[A,A]}$. But I could not.
Please give me a hint on how to solve this problem