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 Th37: :: (7.2.2)
  R = [:A,B:] implies R.:^X = B
proof
  assume
A1: R = [:A,B:];
  thus R.:^X c= B;
  thus B c= R.:^X
  proof
    let a be object;
    assume
A2: a in B;
    then reconsider B as non empty set;
    reconsider a as Element of B by A2;
    for x being set st x in X holds a in Im(R,x)
    proof
      let x be set;
      assume x in X;
      then [x,a] in R by A1,ZFMISC_1:87;
      hence thesis by Th9;
    end;
    hence thesis by Th25;
  end;
end;
