I'm having some difficulty understanding some material in a book I'm reading, Introduction to Set Theory 3rd ed. by Hrbacek and Jech.
The question is about a proof of a theorem given on page 118 in the book.
The theorem that I'm having difficult with is Theorem 3.6 (given on page 113):
Let G be an operation.For any set a there is a unique infinite sequence ⟨an|n∈N⟩ such that
(b)an+1= G (an,n)for all n∈N
The proof provided in the book on page 118 is as follows:
Let G be an operation. We want to find, for every set a, a sequence⟨an|n∈N⟩ such that a0 = a and an+1= G(an,n) for all n∈N.
By the parametric version of the Transfinite Recursion Theorem 4.11, there is an operation F such that F(0) = a and F(n+1)=G(F(n),n) for all n∈N.
Now we apply the Axiom of Replacement: There exists a sequence ⟨an|n∈N⟩that is equal to F↾ω and the Theorem follows.
The parametric version of the Transfinite Recursion Theorem 4.11 is as follows:
Given binary operations G1,G2, and G3 there is an operation F such that for all z
F(z,0) = G1(z,0),
F(z,α+1) = G2(z,Fz(α)) for all ordinals α, and
F(z,α) = G3(z,Fz↾α)for all limit ordinals α.
Now, I understand everything in the proof of Theorem 3.6 except how the parametric version of Theorem 4.11 is used to derive the operation F in the proof. Can can someone please help me fill in blanks?