Bijection on a component of a cartesian product

I have been recently studying relations and mappings and I have come across the following problem. Consider two non empty finite sets $I,J$ and their cartesian product $I\times J$. Let $f\colon I\times J\to I\times J$ be some bijection. Suposse now, that $|I|=n\in\mathbb{N}$ and let $i_1,i_2,\dots,i_n$ be the elements of $I$. For each $1\leq j\leq n$ define $$A_j=\{a\in I; (\exists j_1,j_2\in J)\ [f((i_j,j_1))=(a,j_2)]\}$$ My question is, does now necessarily exists a bijection $g\colon I\to I$ such that $g(i_j)\in A_j$ for every $j=1,2,\dots,n$? I would appreciate any help with this, as I am really stuck. Thanks.

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