Axiom 3.6

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 $z$, $$ z\in\{y:~P(x,y)\text{ is true for some } x \in A\}\iff P(x,z)\text{ is true for some } x\in A. $$

Links to this page

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

Note that the set $f(S)$ is well-defined thanks to the axiom of replacement (Axiom 3.6). One can also define $f(S)$ using the axiom of specification (Axiom 3.5) instead of replacement, but we leave this as a challenge to the reader.
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.