Property of the projection onto Singular Vectors

I come across this property while reading a paper and cannot find it anywhere: The property goes as follow: given a matrix $$A \in R^{N\times n}$$, denote $$U,V$$ as the subspaces spanned by left and right singular vectors of $$A$$. Denote $$P_U$$ as the projection matrix on the subspace $$U$$. We have that$$A = P_U A$$ Furthermore, if $$rank(A) = r$$, denote $$U_r$$ as the subspace spanned by the first $$r$$ left singular vectors, and $$P_{U_r}$$ as the projection matrix to $$U_r$$, we have $$A=P_{U_r} A$$

I would really appreaciate if anyone can point out the name of this property and possibly show a sketch proof if it is simple enough.

Thanks so much.