Is there a universally accepted "nested element of" relation?

Question

Let $f(a)=\{b,\{a\}\}$

Let $f^n$ indicate $n$ compositions of $f$.

Is there a (ideally universally accepted or standard) nested element relation $\prec$ of the obvious meaning such that $\forall n\in\Bbb N:a\prec f^n(a)$? How is this normally treated?

Suppose there exists a limit $Y=\lim_{n\to\infty}f^n(a)$. Does $\prec$ also cover $a\prec Y$?

Answer 1

You could write $a\in\mathrm{TC}(f^n(a))$ where $\mathrm{TC}$ is the transitive closure operation.