In Bavli Brachot 50a-b, Rava compares a rule for a zimmun to a rule for broken/repaired metal utensils.

In particular, he says that in the case of three people, each coming from a separate group, each group which could make a zimmun: the three forming a new group may make a zimmun if and only if their original groups didn't include them in a zimmun.

Rava learns this halakha from the mishna Keilim 18:9, where a metal bed, half of which was stolen or lost, or it was divided by multiple inheritors, is declared insusceptible to tumah; if the bed is subsequently put back together, it is susceptible to tumah from that point on (and doesn't re-acquire any previous tumah).

‮דאמר רב הונא שלשה שבאו משלש חבורות אינן רשאין ליחלק אמר רב חסדא והוא שבאו משלש חבורות של שלשה בני אדם אמר רבא ולא אמרן אלא דלא אקדימו הנך ואזמון עלייהו בדוכתייהו אבל אזמון עלייהו בדוכתייהו פרח זימון מינייהו אמר רבא מנא אמינא לה דתנן מטה שנגנבה חציה או שאבדה חציה או שחלקוה אחין או שותפין טהורה החזירוה מקבלת טומאה מכאן ולהבא מכאן ולהבא אין למפרע לא אלמא כיון דפלגוה פרח לה טומאה מינה הכא נמי כיון דאזמון עלייהו פרח זימון מינייהו:

Rav Huna said: Three individuals who came from three different groups and sat together to continue their meals also form a zimmun and are not permitted to divide. Rav Ḥisda said: And that is only the halakha in a case where the three individuals came from three groups of three people each. Rava said: We only said this halakha in a case where those members of the previous groups did not include them in the zimmun in their original place, but in a case where they included them in the zimmun in their original place, their obligation to participate in a zimmun has left them. Rava said: From where do I derive to say this halakha? As we learned in a mishna: A ritually impure bed, half of which was stolen or half of which was lost, or it was divided by brothers after they inherited it from their father, or was divided by partners, it is ritually pure. However, if they restored it and reattached the parts, it is susceptible to ritual impurity from here on. Rava infers: From here on, yes, retroactively, no. Apparently, once they divided it, the ritual impurity left it. Here, too, once they included them in the zimmun, their obligation left them and they do not reassume their previous obligation. [translation from Sefaria, with some explanations cut out]

Does Rava really learn this halakha from the mishna, or is it just a comparison (as a memory aide)? Why does Rava pick this particular mishna (as his source)? As the (cut-out) explanation on Sefaria says about the metal bed: "This is true with regard to any ritually impure utensil that was broken or divided; it is no longer a utensil and is therefore ritually pure." Indeed, the idea of tumah returning upon building something out of broken pieces only applies to metal utensils, so wouldn't a comparison to a non-metal utensil be more effective? Is there a deeper connection that I'm missing?

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