probability density of a probability density of a probability density ... limiting distribution?

Question

probability density of a probability density of a probability density ... limiting distribution? If we start with a truncated $p(x)=N(0,1)$, with, say cutoffs at +/- 5, and compute the density $d_1(p_1)$ of the probability density $p(x)$ ... $$d_1(p_1)=\int_{-\infty}^{\infty} \delta(p_1-p(x))dx$$ and then iterate that $$d_2(p_2)=\int_{-\infty}^{\infty} \delta(p_2-d_1(x))dx$$ would all initial distributions $p(x)$ converge to the same recursive metadensity $d_n(p_n)$ in the limit that $n\rightarrow \infty$?