assume A3:
the carrier of X <> {}
; :: thesis: { x where x is Element of X : x ref = rng A } is List of X

{ x where x is Element of X : x ref = rng A } c= the carrier of X

{ x where x is Element of X : x ref = rng A } c= the carrier of X

proof

hence
{ x where x is Element of X : x ref = rng A } is List of X
; :: thesis: verum
let z be object ; :: according to TARSKI:def 3 :: thesis: ( not z in { x where x is Element of X : x ref = rng A } or z in the carrier of X )

assume z in { x where x is Element of X : x ref = rng A } ; :: thesis: z in the carrier of X

then ex x being Element of X st

( z = x & x ref = rng A ) ;

hence z in the carrier of X by A3; :: thesis: verum

end;assume z in { x where x is Element of X : x ref = rng A } ; :: thesis: z in the carrier of X

then ex x being Element of X st

( z = x & x ref = rng A ) ;

hence z in the carrier of X by A3; :: thesis: verum