reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;
reserve D for non empty set;
reserve A,B for non empty set;

theorem
  for f being Function, A0, C being set st C c= A0 holds f.:C = (f|A0).: C
proof
  let f be Function, A0, C be set;
  assume
A1: C c= A0;
  thus (f|A0).:C = (f*(id A0)).:C by RELAT_1:65
    .= f.:((id A0).:C) by RELAT_1:126
    .= f.:C by A1,FUNCT_1:92;
end;
