I am learning set theory based on Pinter textbook. In the textbook, the author suggests that the axiom of replacement implies the axiom of pairing and the axiom of subset. I was trying to deduce the axiom of subset based on the axiom of replacement and some other axioms. But I noticed that I don't even need the axiom of replacement to do it. This is my claim. For any set, the axiom of power set guarantees the existence of power set. Then, for a subset of the set, the subset is a member of the power set, so it is a set, due to the definition of set.
If this is true, why do we even need the axiom of power set? What is wrong? And how should I properly deduce the axiom of subset from the axiom of replacement?