You can think of "fi" as shorthand for the list of individual functions "f1, f2, ..., fn, ..." Understood as such, the correct sentence would be:
Among the functions fi there are no repetitions.
This would be a shorthand for:
Among the functions f1, f2, ..., fn, ... there are no repetitions.
Notice the plural "functions" and the use of the plural verb "are".
Alternatively (and maybe less standardly), you could think of "fi" as a kind of count noun which applies to every function in the list. Understood as such, the correct sentence would be:
Among the fi's there are no repetitions.
Normally, you would not use an apostrophe in a pluralization. But you can get away with using the apostrophe here (even though it is not indicating possession), just like you can get away with using it to make a pluralization for something like a letter, as in:
There are two a's in the word actual.
The apostrophe is needed to forestall confusion.
I don't think there is hard and fast rule about which is the right way to view the mathematical convention of using "fi". Your best bet is to pick one way of understanding it and stick to it. Or else forget about it entirely and just say:
Among the functions there are no repetitions.
After all, why muddy the page with needless formalism?