reserve v,x,x1,x2,x3,x4,y,y1,y2,y3,y4,z,z1,z2 for object,
  X,X1,X2,X3,X4,Y,Y1,Y2,Y3,Y4,Y5,
  Z,Z1,Z2,Z3,Z4,Z5 for set;
reserve p for pair object;
reserve R for Relation;
reserve xx1 for Element of X1,
  xx2 for Element of X2,
  xx3 for Element of X3;
reserve xx4 for Element of X4;
reserve A1 for Subset of X1,
  A2 for Subset of X2,
  A3 for Subset of X3,
  A4 for Subset of X4;
reserve x for Element of [:X1,X2,X3:];
reserve x for Element of [:X1,X2,X3,X4:];
reserve x for object;

theorem Th71:
 for R being non empty Relation, x being object
  holds Im(R,x) = { I`2 where I is Element of R: I`1 = x }
 proof let R be non empty Relation, x being object;
   set X = { I`2 where I is Element of R: I`1 = x };
  thus Im(R,x) c= X
   proof let z be object;
    assume z in Im(R,x);
     then consider y being object such that
A1:   [y,z] in R and
A2:   y in {x} by RELAT_1:def 13;
A3:   y = x by A2,TARSKI:def 1;
     y = [y,z]`1 & z = [y,z]`2;
    hence z in X by A1,A3;
   end;
  let z be object;
  assume z in X;
   then consider I being Element of R such that
A4: z= I`2 and
A5: I`1 = x;
A6: I = [I`1,I`2] by Th15;
   x in {x} by TARSKI:def 1;
  hence z in Im(R,x) by A4,A5,A6,RELAT_1:def 13;
 end;
