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It is very common in set theory to prove that a particular model or structure is well-founded by mapping it into a fixed well-founded structure. The point is that if $j:\langle M,{\in^M}\rangle\to \langle N,{\in}\rangle$ is $\in$-preserving, then any instance of ill-foundedness in $M$ would carrry over to $N$, which has none; and so $M$ is well-founded. This method is often applied in the context of iterated ultrapower constructions.