Finitary symmetric group is automorphism-faithful in symmetric group

Statement
Let $$S$$ be any set. Then, the finitary symmetric group $$\operatorname{FSym}(S)$$ is an automorphism-faithful subgroup in $$\operatorname{Sym}(S)$$: any nontrivial automorphism of $$\operatorname{Sym}(S)$$ that restricts to an automorphism on $$\operatorname{FSym}(S)$$ restricts to a nontrivial automorphism on $$\operatorname{FSym}(S)$$.

Related facts

 * Finitary symmetric group is centralizer-free in symmetric group
 * Finitary symmetric group is characteristic in symmetric group
 * Symmetric groups on infinite sets are complete
 * Transposition-preserving automorphism of finitary symmetric group is induced by conjugation by a permutation

Facts used

 * 1) uses::Finitary symmetric group is centralizer-free in symmetric group
 * 2) uses::Finitary symmetric group is normal in symmetric group
 * 3) uses::Normal and centralizer-free implies automorphism-faithful

Proof
The proof follows directly by piecing together facts (1)-(3).