How to transform plane equation

Question

I have the following mapping from R3 -> R3:

Vector3 f(Vector3 v) {
  // Only points with |x| < z and |y| < z and 0 < z <= SomeConstant are of interest
  assert(abs(v.x) <= v.z);
  assert(abs(v.y) <= v.z);
  assert(v.z > 0);
  assert(v.z <= SomeConstant);
  Vector3 res;
  res.x = v.x / v.z;
  res.y = v.y / v.z;
  res.z = v.z / SomeConstant;
  return res;
}

under this mapping, points that lie on a plane will also still lie on a plane afterwards.

Now, if I have a plane equation in the form a*x + b*y + c*z - d = 0, how can I transform the parameters P = [a, b, c, d] so that I'm getting a plane P' = [a', b', c', d'] such that all points on the original plane P will lie on plane P' after being transformed by the above function?