Transitive collapse fixes transitive subsets
= Transitive collapse fixes transitive subsets
If $X$ has a well-founded extensional inherited membership relation and $T\subseteq X$ is a <transitive set>, its <Mostowski collapse theorem> map $\pi$ fixes every member of $T$. The recursion $\pi(x)=\{\pi(y):y\in x\cap X\}$ and <well-founded relation> induction give this pointwise. If also $T\in X$, then $\pi(T)=T$.