Add/removing classes depending on existing classes.

Add/removing classes depending on existing classes.

Hi Folks

Basically we have one large image with four thumbnails sitting below (each 25% in width).
I've been able to swap out the large images when the corresponding thumbs are clicked.
My issue is trying to add/remove the classes 'media-type-picture' and 'media-type-video' to/from #big-image depending on the class of the anchor tag being clicked (.picture or .video).
I'm a noob, so please be as gentle as possible.
Many Thanks

The markup:

<div>
      <h3>Kenya portfolio</h3>
      <p id="big-image"><a href=""><img src="big-images/image-1.jpg"></a></p>
      <ul class="thumbs clearfix">
           <li><a href="big-images/image-1.jpg" class="picture"><img src="thumbnails/thumb-1.jpg"></a></li>
           <li><a href="big-images/image-2.jpg" class="video"><img src="thumbnails/thumb-2.jpg"></a></li>
           <li><a href="big-images/image-3.jpg" class="video"><img src="thumbnails/thumb-3.jpg"></a></li>
           <li><a href="big-images/image-4.jpg" class="picture"><img src="thumbnails/thumb-4.jpg"></a></li>
      </ul> 
</div>

The relevant css:

.media-type-picture,
.media-type-video {
code for a transparent overlay 
containing a web icon...
}
.media-type-picture:before,
.media-type-video:before {
font-family: 'icomoon';
etc
}
.media-type-picture:before {content: "\e602";}
.media-type-video:before {content: "\e603";}

The jQuery:

This first bit works to initially swap out the corresponding images

$(function() 
{
    $(".thumbs li a").click(function(e){
        var href = $(this).attr("href");
        $("#big-image a img").attr("src", href);
        e.preventDefault();
        return false;
    });
});


The following is where I have ended up... (after much hair tearing!!)

$(function(){

    $('.thumbs li a').click(function(e){
      var bigImage = $('#big-image');
      var firstImage = $('bigImage').addClass('media-type-picture');

      if ($(this).hasClass('picture')){
          bigImage.removeClass(' ');
          bigImage.addClass('media-type-picture');
      } else {
         bigImage.removeClass(' ');
         bigImage.addClass('media-type-video');
      }
        e.preventDefault();
        return false;
    });
});