theorem
  x in X implies Im([:X,Y:],x) = Y
  proof assume
A1: x in X;
    thus Im([:X,Y:],x) c= Y
    proof let y be object;
      assume y in Im([:X,Y:],x);
      then ex z st [z,y] in [:X,Y:] & z in {x} by Def11;
      hence y in Y by ZFMISC_1:87;
    end;
    let y be object;
    assume y in Y; then
A2: [x,y] in [:X,Y:] by A1,ZFMISC_1:87;
    x in {x} by TARSKI:def 1;
    hence y in Im([:X,Y:],x) by A2,Def11;
  end;
