Suppose that $R\subseteq \mathcal{P}(\mathbb{N})\times\mathcal{P}(\mathbb{N})$ is a relation such that $(x,y)\in R$ only if $|x|=|y|$.
Say that a partition $P$ divides a set $x$ if $x$ is the union of some subset of $P$.
Suppose that for every $(x,y)\in R$, and finite partition $P$ of $\mathbb{N}$ that divides $x$ there is (i) a partition $Q$ of some subset of $\mathbb{N}$ that divides $y$ and (ii) some bijection $f:P\to Q$ such that $(\bigcup X,\bigcup f(X))\in R$ whenever $X\subseteq P$ and such that $(\bigcup X,\bigcup f(X))=(x,y)$ for some $X\subseteq P$.
My question is: does it follow that there is a subset $X$ of $\mathcal{P}(\mathbb{N})$ and order isomorphism $h: \mathcal{P}(\mathbb{N})\to X$ preserving cardinality such that $h\subseteq R$?
(Another question I'm also curious about: clearly if $R$ is a union of such isomorphisms, it satisfies the conditions of the question. I'd like to know whether any $R$ satisfying those conditions can be expressed as a union of such isomorphisms (it clearly can't if the answer to the first question is negative).)