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 Th43:
  for X, Y being set holds
  X meets R~.:Y iff ex x,y being set st x in X & y in Y & x in Im(R~,y)
proof
  let X, Y be set;
  hereby
    assume X meets R~.:Y;
    then consider a being object such that
A1: a in X and
A2: a in R~.:Y by XBOOLE_0:3;
    consider b being object such that
A3: [b,a] in R~ and
A4: b in Y by A2,RELAT_1:def 13;
A5: b in {b} by TARSKI:def 1;
     reconsider a,b as set by TARSKI:1;
    take a,b;
    thus a in X by A1;
    thus b in Y by A4;
    thus a in Im(R~,b) by A3,A5,RELAT_1:def 13;
  end;
  given x,y being set such that
A6: x in X and
A7: y in Y and
A8: x in Im(R~,y);
  ex a being object st ( [a,x] in R~)&( a in {y}) by A8,RELAT_1:def 13;
  then [y,x] in R~ by TARSKI:def 1;
  then x in R~.:Y by A7,RELAT_1:def 13;
  hence thesis by A6,XBOOLE_0:3;
end;
