let T, S be 1-sorted ; :: thesis: for f being Function of T,S st rng f = [#] S & f is one-to-one holds
( (f ") * f = id (dom f) & f * (f ") = id (rng f) )
let f be Function of T,S; :: thesis: ( rng f = [#] S & f is one-to-one implies ( (f ") * f = id (dom f) & f * (f ") = id (rng f) ) )
assume that
A1: rng f = [#] S and
A2: f is one-to-one ; :: thesis: ( (f ") * f = id (dom f) & f * (f ") = id (rng f) )
A3: f is onto by A1, FUNCT_2:def 3;
( (f ") * f = id (dom f) & f * (f ") = id (rng f) ) by A2, FUNCT_1:39;
hence ( (f ") * f = id (dom f) & f * (f ") = id (rng f) ) by A2, A3, Def4; :: thesis: verum