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The property is listed on AirBnB.com, where you get the address and the name of the person operating the AirBnB.
They don't need an elaborate sting operation. They just take the link, test book it through the website, confirm the address and name of the operator from the booking, and they have all the evidence they need. Hit up AirBnB for a refund (illegal rental) and it doesn't even cost them anything. Hour or two of work at most.