Is the languague L={, M accepts a finite amount of words} decdidable?

Is $ L=\{<M> | L(M) \ is \ finite\} $ decidable ? M is a TM.

I think its relative simple to proof with the theorem of rice. But I am interested in a solution which does not use the Rice theorem.

This my try : Let f(<m,w>) be a function which works in the following way :

  1. Run w on M
  2. If M accepts Construct a TM Mwhich accepts only the word w and return M
  3. If M rejects Construct a TM Mwhich accepts everything. Return M

So if m is in $ A_{TM}= \{<M,w>|M \ accepts \ w\}$ we know that f(<m,w>) is in L. If m is not in A then we know that f(<m,w>) does accept every word and therefore infinity words. So f(<m,w>) not in L.

Is this a correct mapping reduction ?