wpf - DataGrid をコンボボックスの表示値で並べ替える

Question

親子関係を持つ 2 つの DataGrid があり、両方ともデータ コレクションにバインドされています。子 DataGrid は、ID の代わりに親の名前を表示するために DataGridComboBoxColumn を実装します。

問題:How to sort second DataGrid by this column, but by Parent's Name, not by Id?

XAML:

<Window x:Class="WpfApplication1.MainWindow"
    xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
    xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
    xmlns:local="clr-namespace:WpfApplication1"
    x:Name="MainWnd"
    Title="MainWindow" Height="500" Width="650" Loaded="MainWindow_OnLoaded">
<Grid>
    <Grid.RowDefinitions>
        <RowDefinition Height="*"/>
        <RowDefinition Height="Auto"/>
        <RowDefinition Height="2*"/>
    </Grid.RowDefinitions>

    <GridSplitter Grid.Row="1" VerticalAlignment="Center" HorizontalAlignment="Stretch" Height="5"/>

    <local:BaseGrid Grid.Row="0" Margin="3" AutoGenerateColumns="False"
                    ItemsSource="{Binding ElementName=MainWnd, Path=AColView.View}">
        <DataGrid.Columns>
            <DataGridTextColumn Header="Id" Binding="{Binding Path=Id}"/>
            <DataGridTextColumn Header="Name" Binding="{Binding Path=Aname}"/>
        </DataGrid.Columns>
    </local:BaseGrid>

    <local:BaseGrid Grid.Row="2" Margin="3" AutoGenerateColumns="False"
                    ItemsSource="{Binding ElementName=MainWnd, Path=BColView.View}">
        <DataGrid.Columns>
            <DataGridTextColumn Binding="{Binding Path=Id}">
                <DataGridTextColumn.Header>
                    <local:CaptionExtender Caption="Id"/>
                </DataGridTextColumn.Header>
            </DataGridTextColumn>
            <DataGridTextColumn Binding="{Binding Path=Bname}">
                <DataGridTextColumn.Header>
                    <local:CaptionExtender Caption="Name"/>
                </DataGridTextColumn.Header>
            </DataGridTextColumn>
            <DataGridComboBoxColumn x:Name="BparentColumn"
                                    SelectedValueBinding="{Binding ParentId}"
                                    SelectedValuePath="Id"
                                    DisplayMemberPath="Aname">
                <DataGridComboBoxColumn.Header>
                    <local:CaptionExtender Caption="Parent"/>
                </DataGridComboBoxColumn.Header>
            </DataGridComboBoxColumn>

        </DataGrid.Columns>
    </local:BaseGrid>
</Grid>
</Window>

コードビハインド:

public partial class MainWindow
{
    private ObservableCollection<A> ACol { get; set; }
    private ObservableCollection<B> BCol { get; set; }

    public CollectionViewSource AColView { get; private set; }
    public CollectionViewSource BColView { get; private set; }

    public MainWindow()
    {
        ACol = new ObservableCollection<A>();
        BCol = new ObservableCollection<B>();

        AColView = new CollectionViewSource { Source = ACol };
        BColView = new CollectionViewSource { Source = BCol };

        InitializeComponent();

        BparentColumn.ItemsSource = ACol;

    }

    private void MainWindow_OnLoaded(object sender, RoutedEventArgs e)
    {
        // add parents
        for (int i = 1; i <= 5; ++i)
            ACol.Add(new A { Id = i, Aname = "Parent " + i.ToString(CultureInfo.InvariantCulture) });

        // add children
        for (int i = 1; i <= 1000; ++i)
            BCol.Add(new B { Id = i, Bname = "Child " + i.ToString(CultureInfo.InvariantCulture), ParentId = (i % ACol.Count) + 1 });
    }
}

public class A
{
    public int Id { get; set; }
    public string Aname { get; set; }
}

public class B
{
    public int Id { get; set; }
    public int ParentId { get; set; }
    public string Bname { get; set; }
}

これが全体のソリューションです

これが私が達成したいことです: