Keilim, Chapter 11, Mishnah 5. Our mishnah turns to the bridle of a horse, meaning the whole assembly of gear that sits on the animal's head, and asks which of its pieces count as a keli that can receive tumah.
Before entering the words of the mishnah we should have the bridle clearly in mind, piece by piece. At its center sits the bit, the metal bar placed inside the animal's mouth. Fastened to it are the reins, the long straps held in the rider's hands, through which he directs the horse where to go. Further straps serve to keep this apparatus steady, holding each of those two parts in its proper place on the animal's head. And some bridles carry extra straps that run down along the horse's cheeks, straps that accomplish nothing practical whatsoever; they are there for decoration alone.
When all these pieces are joined together into a single working bridle, the bridle as a whole is a utensil and the entire assembly is able to become tamei. The discussion of our mishnah concerns a bridle made entirely of metal, and specifically the situation in which the various parts are not fastened to one another. Each component is lying on its own, detached from the rest. The question is which pieces still have enough of an independent identity to be considered a keli in their own right.
The mishnah begins: "Akrav shel perumbia temei'ah" - the bit of the horse's bridle is susceptible to tumah. Even when it is not attached to the remainder of the bridle, the bit standing alone retains its own identity as a functional item, and therefore it can become tamei by itself.
"Vehalechayayim tehorim" - the cheek pieces are tahor. Those decorative straps that sometimes run along the horse's cheeks are automatically pure. Since they are nothing but ornamentation and fill no practical role whatsoever, they do not have the status of a utensil, and so they are not capable of receiving tumah.
"Rabbi Eliezer metamei balechayayim" - Rabbi Eliezer disagrees and rules that the cheek pieces can become tamei. In his view, once these straps are mounted on the bridle they do contribute something useful to it, and therefore they carry that status of susceptibility with them even when they are detached and lying alone. Rabbi Eliezer agrees with the principle that a purely ornamental object cannot become tamei, but he limits it: that rule applies only to something whose entire existence is decorative. Once an item is attached to a keli and plays some role together with it, it acquires the capacity to become tamei, and it keeps that capacity even on its own. Hence, according to Rabbi Eliezer, the cheek straps are susceptible.
Now the mishnah presents the view of the Sages: "VaChachamim omrim", and the Chachamim say, "ein tamei ela akrav, uvish'at chiburan hakol tamei". By their ruling, of the entire headgear only one piece carries susceptibility to tumah while it stands by itself, and that is the bar which enters the animal's mouth.
Read quickly, this final opinion might seem identical to the mishnah's opening statement, since neither one grants susceptibility to the decorative straps at the cheeks. The real dispute, however, concerns everything else in the assembly. The Tanna Kamma excluded those straps and nothing further: every remaining piece keeps its own standing as a keli even while it lies separated. The Chachamim narrow the list dramatically, holding that among the detached parts the akrav alone is susceptible, while the rest, reins and holding straps alike, are tahor.
That is the force of their closing words, "uvish'at chiburan hakol tamei": it is only while the bridle is assembled and all its pieces are joined together that the remaining components can become tamei. Once they are taken apart, in the view of the Chachamim, the bit alone stands as a keli in its own right.