%------------------------------------------------------------------------------
% File     : SEV604^1 : TPTP v9.3.1. Released v9.3.0.
% Domain   : Set Theory
% Problem  : Set vimage singleton
% Version  : Especial.
% English  :

% Refs     :
% Source   : [TPTP]
% Names    : 001 [TPTP]

% Status   : Theorem
% Rating   : 0.00 v9.3.0
% Syntax   : Number of formulae    :   15 (   4 unt;  11 typ;   0 def)
%            Number of atoms       :   11 (   5 equ;   0 cnn)
%            Maximal formula atoms :    1 (   2 avg)
%            Number of connectives :   47 (   1   ~;   1   |;   0   &;  45   @)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    7 (   5 avg)
%            Number of types       :    3 (   2 usr)
%            Number of type decls  :   11 (   4 !>P;   0 !>D)
%            Number of type conns  :   11 (  11   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   10 (   9 usr;   2 con; 0-4 aty)
%            Number of variables   :   16 (   0   ^;  11   !;   0   ?;  16   :)
%                                         (   5  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TH1_THM_EQU_NAR_NDT

% Comments : Originally from Isabelle 2015
%------------------------------------------------------------------------------
thf(ty_set,type,
    set: $tType > $tType ).

thf(ty_itself,type,
    itself: $tType > $tType ).

thf(ty_a,type,
    a: $tType ).

thf(ty_b,type,
    b: $tType ).

thf(c_bot,type,
    bot: 
      !>[A: $tType] : A ).

%thf(c_type,type,(
%    !>[A: $tType] :
%      ( itself @ A ) )).

thf(c_insert,type,
    insert: 
      !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).

thf(c_vimage,type,
    vimage: 
      !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).

thf(c_member,type,
    member: 
      !>[A: $tType] : ( A > ( set @ A ) > $o ) ).

thf(c_a,type,
    a2: a ).

thf(c_b,type,
    b2: b ).

thf(c_f,type,
    f: a > b ).

thf(vimage_eq,axiom,
    ! [A: $tType,B: $tType,A2: A,F: A > B,B2: set @ B] :
      ( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ B2 ) )
      = ( member @ B @ ( F @ A2 ) @ B2 ) ) ).

thf(empty_iff,axiom,
    ! [A: $tType,C: A] :
      ~ ( member @ A @ C @ ( bot @ ( set @ A ) ) ) ).

thf(insert_iff,axiom,
    ! [A: $tType,A2: A,B3: A,A3: set @ A] :
      ( ( member @ A @ A2 @ ( insert @ A @ B3 @ A3 ) )
      = ( ( A2 = B3 )
        | ( member @ A @ A2 @ A3 ) ) ) ).

thf(vimage_singleton_eq,conjecture,
    ( ( member @ a @ a2 @ ( vimage @ a @ b @ f @ ( insert @ b @ b2 @ ( bot @ ( set @ b ) ) ) ) )
    = ( ( f @ a2 )
      = b2 ) ) ).

%------------------------------------------------------------------------------