Using filters to reduce bandwidth of received signal
I am talking about the modulated carrier, and the 5-25 KHz is the
  bandwidth of the FM signal received.


If you are talking about applying a bandpass filter to the modulated carrier then no, this will not restrict the bandwidth of your demodulated signal. That's not how FM works. The amplitude of the modulating signal determines how far off-centre the carrier is pushed.

It sounds like your unmodulated carrier is 15 kHz and the modulating signal pushes this up to 25 kHz and down to 5kHz - quote "5-25 KHz is the bandwidth of the FM signal received". Applying a bandpass filter to this will reduce the amplitudes of the frequencies at 5 kHz and at 25 kHz but any half decent FM demodulator will still convert these reduced levels into the appropriate demodulated signal.