reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;

theorem Th67:
  f"(Y1 /\ Y2) = f"Y1 /\ f"Y2
proof
  for x being object holds x in f"(Y1 /\ Y2) iff x in f"Y1 /\ f"Y2
  proof let x be object;
    reconsider x as set by TARSKI:1;
A1: x in f"Y2 iff f.x in Y2 & x in dom f by Def7;
A2: x in f"(Y1 /\ Y2) iff f.x in Y1 /\ Y2 & x in dom f by Def7;
    x in f"Y1 iff f.x in Y1 & x in dom f by Def7;
    then x in f"(Y1 /\ Y2) iff x in f"Y1 /\ f"Y2 by A1,A2,XBOOLE_0:def 4;
    hence thesis;
  end;
  hence thesis by TARSKI:2;
end;
