jquery mousedown to run once on selected element but remain for use by other elements

I have a '#tsTable tbody' to which I programatically attach tr and td elements, with one column of the td elements each having the same class ('name').

I have a function which watches for a mousedown event on any element with class ('name'), as follows:

$('#tsTable tbody').on('mousedown', '.name, function(event) {
  var currentName = $(this).text();
  $(this).attr('name', currentName);//store name for later
});

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I need to stop this mousedown event being triggered more than once for the same element to protect against accidental multiple clicks by the user.  I tried to do this by using unbind() as follows:

$('#tsTable tbody').on('mousedown', '.name, function(event) {
  var currentName = $(this).text();
  $(this).attr('name', currentName);//store name for later
  $(this).unbind(event);
});

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However, this results in the function being unloaded not only for the first element which is clicked, but also for every other element of the same class ('name'), which is not what I want.

As further explanation, I want to do this:
1. run the mousedown function once only on the clicked element (and ignore further clicks of that same element);
2. then do some other processing with the captured 'currentName' (and other things);
3. then after that processing, reactivate the first mousedown function so it can be reused on the page again (on a different clicked element).
ShanghaiDAsked:
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X-treemCommented:
check on which element the event is triggered and stop the event propagation (bubbling) when required
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ShanghaiDAuthor Commented:
Yes, I'll accept this as the solution.  In fact I can do this by first testing for the existence of the attribute value which is stored on the first mousedown (only) as follows:

$('#tsTable tbody').on('mousedown', '.name, function(event) {
  if($(this).attr('name') == undefined) {
    var currentName = $(this).text();
    $(this).attr('name', currentName);//store name for later
  }
});
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