let f be Function; :: thesis: ( f is one-to-one implies ( dom ((f " ) * f) = dom f & rng ((f " ) * f) = dom f ) )
assume A1: f is one-to-one ; :: thesis: ( dom ((f " ) * f) = dom f & rng ((f " ) * f) = dom f )
then A2: rng f = dom (f " ) by Th55;
then rng ((f " ) * f) = rng (f " ) by RELAT_1:47;
hence ( dom ((f " ) * f) = dom f & rng ((f " ) * f) = dom f ) by A1, A2, Th55, RELAT_1:46; :: thesis: verum