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 Th1:
  [x,y] in f iff x in dom f & y = f.x
proof
  thus [x,y] in f implies x in dom f & y = f.x
  proof
    assume
A1: [x,y] in f;
    hence
A2:   x in dom f by XTUPLE_0:def 12;
     reconsider y as set by TARSKI:1;
     y = f.x by A1,Def2,A2;
    hence thesis;
  end;
  thus thesis by Def2;
end;
