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 Th73:
  (for y st y in rng f ex x st f"{y} = {x}) iff f is one-to-one
proof
  thus (for y st y in rng f ex x st f"{y} = {x}) implies f is one-to-one
  proof
    assume
A1: for y st y in rng f ex x st f"{y} = {x};
    let x1,x2;
    assume that
A2: x1 in dom f and
A3: x2 in dom f;
    f.x1 in rng f by A2,Def3;
    then consider y1 such that
A4: f"{f.x1} = {y1} by A1;
    f.x2 in rng f by A3,Def3;
    then consider y2 such that
A5: f"{f.x2} = {y2} by A1;
    f.x1 in {f.x1} by TARSKI:def 1;
    then x1 in {y1} by A2,A4,Def7;
    then
A6: y1 = x1 by TARSKI:def 1;
    f.x2 in {f.x2} by TARSKI:def 1;
    then x2 in {y2} by A3,A5,Def7;
    hence thesis by A4,A5,A6,TARSKI:def 1;
  end;
  assume
A7: f is one-to-one;
  let y;
  assume y in rng f;
  then consider x being object such that
A8: x in dom f & y = f.x by Def3;
  take x;
  for z being object holds z in f"{y} iff z = x
  proof
    let z be object;
    thus z in f"{y} implies z = x
    proof
      assume
A9:   z in f"{y};
      then f.z in {y} by Def7;
      then
A10:  f.z = y by TARSKI:def 1;
      z in dom f by A9,Def7;
      hence thesis by A7,A8,A10;
    end;
    y in {y} by TARSKI:def 1;
    hence thesis by A8,Def7;
  end;
  hence thesis by TARSKI:def 1;
end;
