  We prove that the Shilov boundary is invariant under proper holomorphic
mappings between some classes of domains (containing among others
quasi-balanced domains with the continuous Minkowski functionals). Moreover, we
obtain an extension theorem for proper holomorphic mappings between
quasi-circular domains.
  Using these results we show that there are no non-trivial proper holomorphic
self-mappings in the tetrablock. Another important result of our work is a
description of Shilov boundaries of a large class of domains (containing among
other the symmetrized polydisc and the tetrablock).
  It is also shown that the tetrablock is not $\mathbb C$-convex.
