Hi I'm having trouble with this proof. I'm not sure if I did the first part right and how I should use the second part properly. I am following the hint in the book to use the Axiom of Replacement to prove step three below and follow through with Replacement and Union to finalize the proof.
1) Prove that the Cartesian Product of two sets is a set
$ A \times B = \left \{ \left \langle u, v \right \rangle \mid u \in A \land v \in B \right \} $
2) Taking the Axiom of Replacement
$ \forall u \forall v \forall w \left ( \psi \left ( u, v \right ) \land \psi \left ( u, w \right ) \rightarrow v = w \right ) \rightarrow \forall z \exists y \forall v \left ( v \in y \leftrightarrow \left ( \exists u \in z \right ) \psi \left ( u, v \right ) \right ) $
3) First prove that the following is a set
$ \forall x \left ( \lbrace \langle x, y \rangle \mid y \in t \rbrace \right ) $
4) Taking (1) from Section 6
$ \langle x, y \rangle = \langle u, v \rangle \iff x = u \land y = v $
5) Use then this u and v for the Axiom of Replacement reduced to the following where v is unique
$ \forall z \exists y \forall v \left ( v \in y \leftrightarrow \left ( \exists u \in z \right ) \psi \left ( u, v \right ) \right ) $
6) $ \forall t \left ( t \in y \right ) $ and $ z = \langle x, y \rangle $
$ \forall \langle x, y \rangle \exists t \forall v \left ( v \in t \leftrightarrow \left ( \exists u \in \langle x, y \rangle \right ) \langle u, v \rangle \right ) $
Hence $ y \in t $ proving that $ \forall x \left ( \lbrace \langle x, y \rangle \mid y \in t \rbrace \right ) $ is a set by the Axiom of Replacement
7) Now taking the Axiom of Union
$ A \cup B = \lbrace x \mid x \in A \lor x \in B \rbrace $
8) And the Axiom of Replacement
$ \forall u \forall v \forall w \left ( \psi \left ( u, v \right ) \land \psi \left ( u, w \right ) \rightarrow v = w \right ) \rightarrow \forall z \exists y \forall v \left ( v \in y \leftrightarrow \left ( \exists u \in z \right ) \psi \left ( u, v \right ) \right ) $
9) Prove that the Cartesian Product of two sets is a set
$ A \times B = \left \{ \left \langle u, v \right \rangle \mid u \in A \land v \in B \right \} $
If there is something I can make more clear that would allow others to more easily aid me let me know. :|