Axiom 3.6. (Replacement). Let \(A\) be a set. For any object \(x\in A\), and any object \(y\), suppose we have a statement \(P(x,y)\) pertaining to \(x\) and \(y\), such that for each \(x\in A\) there is at most one \(y\) for which \(P(x,y)\) is true. Then there exists a set \(\{y:~P(x,y)\text{ is true for some } x\in A\}\), such that for any object \(z\), $$ z\in\{y:~P(x,y)\text{ is true for some } x \in A\}\iff P(x,z)\text{ is true for some } x\in A. $$
Axiom 3.6
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Exercise 3.4.6
\(\Vdash\) { replacement axiom }\(\vdash\) { replacement axiom }
Prove Lemma 3.4.9. (Hint: start with the set \(\{0,1\}^X\) and apply the replacement axiom, replacing each function \(f\) with the object \(f^{-1}(\{1\})\).)
See also Exercise 3.5.11. - Definition 3.4.1
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Chapter3
Exercise 3.4.6. Prove Lemma 3.4.9. (Hint: start with the set \(\{0,1\}^X\) and apply the replacement axiom, replacing each function \(f\) with the object \(f^{-1}(\{1\})\).)\\ See also Exercise 3.5.11.