Suppose that $M$ and $N$ are $L$-structures, with $L$ a language only containing relational symbols, and suppose that I have an elementary bijective map $j:M\to N$, i.e. a bijection such that for every $L$-formula $\varphi(x_0,\dots,x_n)$ and every $n+1$-tuple $a_0,\dots,a_n$ of elements of $M$, $M\models\varphi(a_0,\dots,a_n)$ if and only if $N\models\varphi(j(a_0),\dots,j(a_n))$.
Is it true that $M\cong N$?
I would think it is, since $(a_0,\dots,a_n)\in R^M$ iff $M\models R(a_0,\dots,a_n)$ iff $N\models R(j(a_0),\dots,j(a_n))$ iff $(j(a_1),\dots,j(a_n))\in R^N$, and I would say that this is all we care about. But I cannot find this result mentioned anywhere, which makes me think that my argument is flawed.
Any hint is appreciated!