Setup:
$(N,\sigma)$ satisfies Peano axioms, i.e. $N \equiv ω$.
Using the subset axiom we define, for each $k \in N$, $F_k = \{x \in N \; | \; \phi_k(x) \}$.
Next we define $D = \{d \in N \; | \; F_d = N\}$ and show that $1 \in D$ and when $d \in D$, so is $d + 1$ (using the fact that $F_d = N$). So $D = N$ and for all $k \in N$, $F_k = N$.
What axioms of ZF set theory come into play here?
The answer can be found by reading the comments, so I will list them here:
I noticed that there is a vote to close this question - that it is unclear what I am asking. Well, without details, I guess it isn't the best question. But, the question did 'collect' three very important axioms from ZF set theory.
I thought that since induction isn't an axiom, it is nothing to make a 'fuss' about. But, it seems that I need to take a refresher course in induction. For example, have you used simultaneous induction lately?
For a review with exercises, see
Induction by Richard Earl Mathematical Institute, Oxford August 2003
I am adding the induction tag to this posting.