reserve X,Y for set, x,y,z for object, i,j,n for natural number;

theorem Th19:
  for I being set, a being object holds pr1(I,{a}) is one-to-one
  proof
    let I be set;
    let a be object;
    set f = pr1(I,{a});
    let x,y be object;
    assume
A1: x in dom f & y in dom f;
    then consider i1,a1 being object such that
A2: i1 in I & a1 in {a} & x = [i1,a1] by ZFMISC_1:def 2;
    consider i2,a2 being object such that
A3: i2 in I & a2 in {a} & y = [i2,a2] by A1,ZFMISC_1:def 2;
    assume f.x = f.y;
    then f.(i1,a1) = f.(i2,a2) by A2,A3;
    then
A4: i1 = f.(i2,a2) = i2 by A2,A3,FUNCT_3:def 4;
    a1 = a = a2 by A2,A3,TARSKI:def 1;
    hence x = y by A2,A3,A4;
  end;
