Associated graded object

Question

I have a question, how do I prove the following proposition? A and B are filtered chain complex , $f: A \rightarrow B $ is a filtered chain map if indicated mapping on the associated graded object of $f: A \rightarrow B $ be quasi-isomorphism then $f: A \rightarrow B $ is quasi-isomorphism.

Answer 1

This question and other direction are true. You can use from Five-lemma and following proposition to proof them. The chain map $f: C \rightarrow C'$ is quasi-isomorphism if and only if homology of mapping cone of f be trivial.