Functional predicate
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The idea of the functional predicate allows mathematicians to extend the concept of function beyond specific sets.
A functional predicate 
is a predicate in two variables (in this case, 
and 
) such that and 
together imply 
. If holds, then we may write 
.
Note that this permits us to speak of "functions" which act on all sets. In set theory, the relation 
, for example, cannot yield a function unless it is confined to a specific domain and range. However, we may speak of the functional predicate 
, which may be applied to any set. This permits us to speak of general functions on sets within the context of classical set theory.
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