Thank you for your post!
I'm going to move this thread from the FileMaker Community Feedback Space (which is specifically for input on the Community itself) to the Discussions Space where you should receive more views and potentially more feedback on this topic!
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What you need to do is keep a return-separated list of the value that have already been selected. This can be used as a multi-key in a relationship with not equal operator to exclude choices via a portal or a value list. It's not clear how the appointments are being selected but this video should get you started to learning dwindling value lists.
Your file does have some basic problems to work out before you try to use a dwindling value list or other method to avoid double booking.
Currently, you appear to "book" a patient by selecting a venue and a date. Your "Time" record links by venue and date and this doesn't seem likely to work. Seems like you need to also link by time as well.
Even then, a patient can only be booked once. Booking them for an event in the future would require overwriting their current link to a previous event.
Thus, it would appear that you also need another table used to book patients to a specific venue, date and time. That table would link patients to the other tables in your solution--though if I were in your shoes, I'd consider using one table for both date and time with a link to venue so that you have one record in this table for every possible time slot for every possible venue.
And with all that and a dwindling list, if you have more than one person doing the booking at the same time, there's still a chance of double booking as two or more users might select the same venue/date/time at the same time. Some error checking will be needed to trap for that error while the admin is still in the process of booking a patient.