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 Th44:
  for X, Y being set holds
  (ex x,y being set st x in X & y in Y & x in Im(R~,y)) iff Y meets R.:X
proof
  let X, Y be set;
  thus (ex x,y being set st x in X & y in Y & x in Im(R~,y)) implies
  Y meets R.:X
  proof
    given x,y being set such that
A1: x in X and
A2: y in Y and
A3: x in Im(R~,y);
    consider a being object such that
A4: [a,x] in R~ and
A5: a in {y} by A3,RELAT_1:def 13;
    a = y by A5,TARSKI:def 1;
    then [x,y] in R by A4,RELAT_1:def 7;
    then y in R.:X by A1,RELAT_1:def 13;
    hence thesis by A2,XBOOLE_0:3;
  end;
  assume Y meets R.:X;
  then consider a being object such that
A6: a in Y and
A7: a in R.:X by XBOOLE_0:3;
  consider b being object such that
A8: [b,a] in R and
A9: b in X by A7,RELAT_1:def 13;
A10: [a,b] in R~ by A8,RELAT_1:def 7;
A11: a in {a} by TARSKI:def 1;
   reconsider a,b as set by TARSKI:1;
  take b,a;
  thus b in X by A9;
  thus a in Y by A6;
  thus thesis by A10,A11,RELAT_1:def 13;
end;
