A name is a further definition of the bridge, it puts it on the map, so to speak. You can also use the article "the" if you say "in my city there is the only bridge, which permits you to cross the river within forty miles" or something like that. Just like when you say "this bridge being the only one in my city".

That means that if you use "the" when you mention something for the first time, it awakens the expectation in the listener that you are going to define it further right away. Even if you end the sentence after "bridge" you have to phrase it differently. "In my city there is only one bridge. It is the only one within forty miles which permits you to cross the river."

In "the only bridge, which" the article establishes a link with the following subordinate sentence, announcing it, so to speak.