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
  (Y|`f).:X = Y /\ f.:X
proof
  for y being object holds y in (Y|`f).:X iff y in Y /\ f.:X
  proof let y be object;
    thus y in (Y|`f).:X implies y in Y /\ f.:X
    proof
      assume y in (Y|`f).:X;
      then consider x being object such that
A1:   x in dom(Y|`f) and
A2:   x in X and
A3:   y = (Y|`f).x by Def6;
A4:   y = f.x by A1,A3,Th52;
      then
A5:   y in Y by A1,Th52;
      x in dom f by A1,Th52;
      then y in f.:X by A2,A4,Def6;
      hence thesis by A5,XBOOLE_0:def 4;
    end;
    assume
A6: y in Y /\ f.:X;
    then y in f.:X by XBOOLE_0:def 4;
    then consider x being object such that
A7: x in dom f and
A8: x in X and
A9: y = f.x by Def6;
    y in Y by A6,XBOOLE_0:def 4;
    then
A10: x in dom(Y|`f) by A7,A9,Th52;
    then (Y|`f).x = f.x by Th52;
    hence thesis by A8,A9,A10,Def6;
  end;
  hence thesis by TARSKI:2;
end;
