I've come across such theorem in Galois theory.
Let $F$ be an algebraic extension of field $K$. $F$ is a splitting field over $K$ of a set of polynomials in $K[x]$ iff for every $K$-monomorphism of fields $σ: F \to N$, where $N$ is any splitting field over $K$ containing $F$, we have $σ(F)=F$.
The sketch of proof of one direction is as follows. Let $u$ be a root of $f_i$, which is in the set of polynomials in $K[x]$. $σ(u)$ is also a root of $f_i$. Since $σ$ is an injection, it must simply permute the roots of $f_i$. $F$ is generated over $K$ by all the roots of all the $f_i$, whence $σ(F)=F$.
My question: is the condition of $N$ being a splitting field necessary? Can we replace $N$ by any extension of $K$ containing $F$? (It seems every step in the proof above has nothing to do with $N$ being a splitting field)