The second line of this answer states
The elements of the poset $Z$ are families $G$ of subsets of $X$, that contain $F$ (the fixed $F$ we start out with, and which has the FIP), so we have $F \subseteq G$ for all $G \in Z$, and such that $G$ has the FIP
Question: I am confused about the fact that we have $F\subseteq G$, because $G$ is a family of subsets of $X$, which I think of as a function, while $F$ is just some subset of $X$.
That is, in few words, how can a set be a subset of a function (I realize a function is a set, but it is a set of ordered pairs. Here $F$ is not a set of ordered pairs)
Question in More Detail
Specifically, at this question ("What is the difference between a family and a set") the answer states that
Strictly speaking, a family is a function $I \to U$, where $I$ is an index set and $U$ is a universe that contains the members of the family.
Using the notation of the first block quotations, then the poset $Z$ should consist of functions $G:I\to X$, but since a function is really a collection of ordered pairs, then we can write $G=\{(i,x)\big\vert i\in I,\ x\in X\}$ (note I am slightly abusing notation in that every element of "$X$" in this definition contains $F$)
I am then unclear about how the claim $F\subseteq G$ can hold, given than $G$ is a set of ordered pairs -- albeit of which the second part contains $F$ -- while $F$ does not have ordered pairs (that is, these seem like different objects to me).
