Five lemma

This is a lemma involving commutative diagrams that can be proved by means of diagram chasing

Let $A_{1}\to A_{2}\to A_{3}\to A_{4}\to A_{5}$ and $B_{1}\to B_{2}\to B_{2}\to B_{4}\to B_{5}$ be exact sequences of homomorphisms. Suppose there are maps $f_{i}:A_{i}\to B_{i}$ such that the diagram of all these maps commutes. Then the following are true:

• If $f_{2}$ and $f_{4}$ are injective and $f_{1}$ is surjective, then $f_{3}$ is injective
• If $f_{2}$ and $f_{4}$ are surjective and $f_{5}$ is injective then $f_{3}$ is surjective
• if $f_{1},f_{2},f_{4},f_{5}$ are isomorphisms, then $f_{3}$ is also an isomorphism