A relation on 2 countable sets

Question

Let $R$ be a relation on two countable sets $A$ and $B$, where $R\subset A\times B$, with the following properties:

1. $\forall a\in A$ the set $\{b\in B: (a,b)\in R\}$ is finite.
2. For any finite set $A_0\subset A$: $$|\{b\in B : \exists a_0\in A_0, \text{such that} \ \ (a_0,b)\in R \}|\leq n|A_0|$$ where $n\in\mathbb{N}$.

I need to show then, that there exist $n$ disjoint sets, $B_1,B_2,\dots, B_n$, where $B_i\subset B\ \ \ \ \ \ \forall \ \ 1 \leq i\leq n$, and there exists $n$ one to one and onto functions $f_1,f_2,\dots, f_n$ such that $f_i:B_i\to A \ \ \ \ \ \ \ \forall \ \ 1 \leq i\leq n $?

Thank you for your help.