reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;
reserve X,X1,X2 for Subset of A;
reserve Y for Subset of B;
reserve R,R1,R2 for Subset of [:A,B:];
reserve FR for Subset-Family of [:A,B:];

theorem Th41:
  y in Im(R`,x) iff not [x,y] in R & x in A & y in B
proof
  thus y in Im(R`,x) implies not [x,y] in R & x in A & y in B
  proof
    assume y in Im(R`,x);
    then ex a being object st ( [a,y] in R`)&( a in {x}) by RELAT_1:def 13;
    then [x,y] in [:A,B:] \ R by TARSKI:def 1;
    hence thesis by XBOOLE_0:def 5,ZFMISC_1:87;
  end;
  assume that
A1: not [x,y] in R and
A2: x in A and
A3: y in B;
A4: x in {x} by TARSKI:def 1;
  [x,y] in [:A,B:] by A2,A3,ZFMISC_1:87;
  then [x,y] in [:A,B:] \ R by A1,XBOOLE_0:def 5;
  hence thesis by A4,RELAT_1:def 13;
end;
