reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem Th104:
  y in R.:X iff ex x st x in dom R & [x,y] in R & x in X
proof
  thus y in R.:X implies ex x st x in dom R & [x,y] in R & x in X
  proof
    assume y in R.:X;
    then consider x such that
A1: [x,y] in R & x in X by Def11;
    take x;
    thus thesis by A1,XTUPLE_0:def 12;
  end;
  thus thesis by Def11;
end;
