Let $\mathcal{A}$ be a non-empty systems of non-empty sets such that there is an injective map $f:\bigcup \mathcal{A}\to \mathcal{A}$ such that $a\in f(a)$ for all $a\in\bigcup\mathcal{A}$. Assuming that $|\mathcal{A}| = |\bigcup \mathcal{A}|$, is there always a bijection $\varphi: \bigcup \mathcal{A}\to \mathcal{A}$ such that $a\in \varphi(a)$ for all $a\in\bigcup\mathcal{A}$. ?



Surjectivity from union of a set system to the set system
No - let $\mathcal{A} = \{\{0,1\}\} \cup \{\{n\} : n\in \omega\}$.


