reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;

theorem
  (for X holds f"(f.:X) c= X) implies f is one-to-one
proof
  assume
A1: for X holds f"(f.:X) c= X;
  given x1,x2 such that
A2: x1 in dom f and
A3: x2 in dom f and
A4: f.x1 = f.x2 & x1 <> x2;
A5: f"(f.:{x1}) c= {x1} by A1;
A6: Im(f,x2) = {f.x2} by A3,Th58;
A7: Im(f,x1) = {f.x1} by A2,Th58;
  f.x1 in rng f by A2,Def3;
  then f"(f.:{x1}) <> {} by A7,Th71;
  then f"(f.:{x1}) = {x1} by A5,ZFMISC_1:33;
  hence contradiction by A1,A4,A7,A6,ZFMISC_1:3;
end;
