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The principle of restricted comprehension states that, for any set \(x\) and predicate \(P\), there exists a set (written \(\{a\in x|P(a)\}\)) whose elements are precisely those elements \(a\) of \(x\) which satisfy \(P(a)\): \(\forall x:\exists y:\forall a:(a\in y\Leftrightarrow a\in x\land P(a))\). This is an axiom schema, as a copy of it applies for every predicate \(P\), and it is implied by the stronger axiom of replacement: take a set \(x\) and a predicate \(P\), then either there is no element of \(x\) satisfying \(P\) (and so the required set is \(\emptyset\)), or there is some \(\tilde{a}\in x\) such that \(P(a)\). In the second case, one forms a relation \(\mathscr{R}\) by \(a\mathscr{R}b\Leftrightarrow a\in x \land ((P(a)\land a=b)\lor(\neg P(a)\land b=\tilde{a}))\) which is functional on \(x\), and the image of each \(a\in x\) is either \(a\) itself (if \(P(a)\) holds) or \(\tilde{a}\) (if \(P(a)\) is false), so the image of \(x\) under \(\mathscr{R}\) is precisely those elements of \(x\) for which \(P\) is true. A stronger version of the principle of restricted comprehension which is not an axiom of set theory (as it leads to inconsistencies) is the principle of unrestricted comprehension, which would allow for sets of the form \(\{x|P(x)\}\).