let T, S be 1-sorted ; :: thesis: for f being Function of T,S
for P1 being Subset of S st rng f = [#] S & f is one-to-one holds
f " P1 = (f " ) .: P1
let f be Function of T,S; :: thesis: for P1 being Subset of S st rng f = [#] S & f is one-to-one holds
f " P1 = (f " ) .: P1
let P1 be Subset of S; :: thesis: ( rng f = [#] S & f is one-to-one implies f " P1 = (f " ) .: P1 )
assume A1: ( rng f = [#] S & f is one-to-one ) ; :: thesis: f " P1 = (f " ) .: P1
then f " P1 = (f " ) .: P1 by FUNCT_1:155;
hence f " P1 = (f " ) .: P1 by A1, Def4; :: thesis: verum