VC-dimension of Cylinders within a Cylinder
You need the sigmoid restriction on $f$ for the VC-dimension to be finite.  Otherwise, you can let $f$ behave like a staircase, with arbitrarily many steps.  Then these staircases can have arbitrarily many intersections.  This allows $n$ ranges to admit $2^n$ different subsets.  

If $f(z)$ is a polynomial, then you can bound the VC-dimension using the degree of the polynomial (combined with the degree of the polynomial (2) describing the disc).  But not sure how to apply this type of result for $\tanh$.