Corollary

If $f$ is an injection of A into B and if $(X_\iota)_{\iota \in I}$ is a family of subsets of A whose index set I is not empty, then $f\left\langle \bigcap_{\iota \in I} X_\iota \right\rangle = \bigcap_{\iota \in I} f\langle X_\iota \rangle$.

For we may write $f = i \circ g$, where $i$ is the canonical injection of $f\langle A \rangle$ into B and $g$ is a bijection of A onto $f\langle A \rangle$. If $h$ denotes the inverse mapping of $g$, we have $f\langle X \rangle = \overset{-1}{h}\langle X \rangle$ for every subset X of A, and we are therefore brought back to Proposition 4.