Quasi isomorphism criterion from a filtered map

By [F1] there is a morphism of spectral sequences, whose component at $(p,q)$ on page one identifies with $H_{p+q}({\mathrm{gr}}_{p}f)$. This is an isomorphism by the hypothesis and [F5], for every $p,q$. Thus the required isomorphism is on an entire page, including every zero graded complex.

The map $H_n(f)$ preserves the homology image filtration: a cycle coming from $F_{p}C$ maps to a cycle coming from $F_{p}D$. On its graded quotient, it sends the class of an actual cycle $c$ to that of $f(c)$. The spectral map does the same on the limiting actual-cycle classes, because it is induced by the filtered chain map. Hence the given strong abutment identifications commute with the maps $H_n(f)$; these are the actual maps required by comparison.

Under the conditional strong-convergence and target hypotheses, apply [F3] to steps 1.1–1.2. It makes every $H_n(f)$ an isomorphism, which is exactly the quasi-isomorphism conclusion. This does not claim that completeness of the complexes by itself establishes those convergence hypotheses.

If instead both chain filtrations are degreewise finite, [F4] supplies canonical actual-cycle abutments, two-sided stationarity and finite homology filtrations. Each such finite target filtration is exhaustive and separated, and its lower quotient tail is constant equal to the target, so its completion map is an isomorphism by [F2]. Thus strong convergence and the finite comparison hypotheses hold, and step 2.1 applies. This argument uses the unrestricted bounded theorem in [F4], so no first-quadrant assumption is silently added; the first-quadrant theorem is its named special case. Finite bounds may vary with degree, repeated terms and a one-step filtration are allowed, and neither branch introduces AC.