I tried to plot integer points that fall into a 2D region that's dynamically changed. So I resort to `Cases` and `Select`, but `Cases` with `Condition` is not working expectedly. Here is the simplifed version of my code and result:

    region[u_] := u <= x + y <= u + 2;
    intPoints = Flatten[Table[{x, y}, {x, 0, 13}, {y, 0, 13}], 1];
    Manipulate[
     Column[{
       {ListPlot[intPoints], 
        RegionPlot[region[u], {x, 0, 13}, {y, 0, 13}]},
       {ListPlot[Cases[intPoints, {x_, y_} /; region[u]]],
        ListPlot[Cases[intPoints, {x_, y_} /; Evaluate[region[u]]]],
        ListPlot[Select[intPoints, region[u] /. {x -> #[[1]], y -> #[[2]]} &]]}}],
     {u, 0, 13}
     ]

[![enter image description here][1]][1]

I don't understand why inside `Cases` the point coordinates ***x*** and ***y*** won't go into ***region*** for the `Condition` to be evaluated. Does it first do the replacement of ***x*** and ***y*** and then do the evaluation? But why the additional `Evaluate` operation still doesn't help?

Could someone elaborate me on this? Thanks.

  [1]: https://i.sstatic.net/rH7N1.png