Axiom of Powers Something I'm failing to understand from Halmos "Naive Set theory" book.

If $\Phi $  is a collection of subsets of a set E (that is, $\Phi$ is a subcollection of $\rho (E)$), then write

First of all I would like to point out that letters Phi and rho aren't the ones used in Halmos book. I do not recognize the letters from the book so I had no option but to pick my own.
$$\rho(E) = \{X:X\subset E\}$$
This is the definition of $\rho (E)$ from the book.
My question is: What is the difference between $\rho (E) $ and $\Phi$?

The set $\rho(E) = \mathcal P (E)$ is the set of all the subsets of $E$, while $\Phi$ is a collection of subsets of $E$ (which may not contain all of them). In other words : $\Phi \subseteq \rho(E)$.
For instance, if $E =\{1,2\}$, then the power set of $E$ is $\rho(E) = \{\varnothing, \{1\}, \{2\}, E\}$ and you can take $\Phi = \{\{2\}, E\}$ as a collection of subsets of $E$ (which is not the whole power set).
