%------------------------------------------------------------------------------
% File     : SEU055+1 : TPTP v8.2.0. Released v3.2.0.
% Domain   : Set theory
% Problem  : Functions and their basic properties, theorem 124
% Version  : [Urb06] axioms : Especial.
% English  :

% Refs     : [Byl90] Bylinski (1990), Functions and Their Basic Properties
%          : [Urb06] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb06]
% Names    : funct_1__t124_funct_1 [Urb06]

% Status   : Theorem
% Rating   : 0.56 v8.2.0, 0.67 v8.1.0, 0.53 v7.5.0, 0.62 v7.4.0, 0.57 v7.3.0, 0.55 v7.1.0, 0.52 v7.0.0, 0.50 v6.4.0, 0.54 v6.2.0, 0.68 v6.1.0, 0.73 v6.0.0, 0.74 v5.5.0, 0.78 v5.4.0, 0.79 v5.3.0, 0.81 v5.2.0, 0.70 v5.1.0, 0.71 v5.0.0, 0.67 v4.1.0, 0.70 v4.0.0, 0.71 v3.7.0, 0.65 v3.5.0, 0.74 v3.4.0, 0.68 v3.3.0, 0.79 v3.2.0
% Syntax   : Number of formulae    :   40 (   9 unt;   0 def)
%            Number of atoms       :   96 (  13 equ)
%            Maximal formula atoms :    7 (   2 avg)
%            Number of connectives :   70 (  14   ~;   1   |;  31   &)
%                                         (   6 <=>;  18  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   4 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    9 (   8 usr;   0 prp; 1-2 aty)
%            Number of functors    :    7 (   7 usr;   1 con; 0-2 aty)
%            Number of variables   :   61 (  50   !;  11   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
%            library, www.mizar.org
%------------------------------------------------------------------------------
fof(antisymmetry_r2_hidden,axiom,
    ! [A,B] :
      ( in(A,B)
     => ~ in(B,A) ) ).

fof(cc1_funct_1,axiom,
    ! [A] :
      ( empty(A)
     => function(A) ) ).

fof(cc1_relat_1,axiom,
    ! [A] :
      ( empty(A)
     => relation(A) ) ).

fof(cc2_funct_1,axiom,
    ! [A] :
      ( ( relation(A)
        & empty(A)
        & function(A) )
     => ( relation(A)
        & function(A)
        & one_to_one(A) ) ) ).

fof(d1_tarski,axiom,
    ! [A,B] :
      ( B = singleton(A)
    <=> ! [C] :
          ( in(C,B)
        <=> C = A ) ) ).

fof(d8_funct_1,axiom,
    ! [A] :
      ( ( relation(A)
        & function(A) )
     => ( one_to_one(A)
      <=> ! [B,C] :
            ( ( in(B,relation_dom(A))
              & in(C,relation_dom(A))
              & apply(A,B) = apply(A,C) )
           => B = C ) ) ) ).

fof(existence_m1_subset_1,axiom,
    ! [A] :
    ? [B] : element(B,A) ).

fof(fc12_relat_1,axiom,
    ( empty(empty_set)
    & relation(empty_set)
    & relation_empty_yielding(empty_set) ) ).

fof(fc1_subset_1,axiom,
    ! [A] : ~ empty(powerset(A)) ).

fof(fc1_xboole_0,axiom,
    empty(empty_set) ).

fof(fc2_subset_1,axiom,
    ! [A] : ~ empty(singleton(A)) ).

fof(fc3_relat_1,axiom,
    ! [A,B] :
      ( ( relation(A)
        & relation(B) )
     => relation(set_difference(A,B)) ) ).

fof(fc4_relat_1,axiom,
    ( empty(empty_set)
    & relation(empty_set) ) ).

fof(fc5_relat_1,axiom,
    ! [A] :
      ( ( ~ empty(A)
        & relation(A) )
     => ~ empty(relation_dom(A)) ) ).

fof(fc7_relat_1,axiom,
    ! [A] :
      ( empty(A)
     => ( empty(relation_dom(A))
        & relation(relation_dom(A)) ) ) ).

fof(rc1_funct_1,axiom,
    ? [A] :
      ( relation(A)
      & function(A) ) ).

fof(rc1_relat_1,axiom,
    ? [A] :
      ( empty(A)
      & relation(A) ) ).

fof(rc1_subset_1,axiom,
    ! [A] :
      ( ~ empty(A)
     => ? [B] :
          ( element(B,powerset(A))
          & ~ empty(B) ) ) ).

fof(rc1_xboole_0,axiom,
    ? [A] : empty(A) ).

fof(rc2_funct_1,axiom,
    ? [A] :
      ( relation(A)
      & empty(A)
      & function(A) ) ).

fof(rc2_relat_1,axiom,
    ? [A] :
      ( ~ empty(A)
      & relation(A) ) ).

fof(rc2_subset_1,axiom,
    ! [A] :
    ? [B] :
      ( element(B,powerset(A))
      & empty(B) ) ).

fof(rc2_xboole_0,axiom,
    ? [A] : ~ empty(A) ).

fof(rc3_funct_1,axiom,
    ? [A] :
      ( relation(A)
      & function(A)
      & one_to_one(A) ) ).

fof(rc3_relat_1,axiom,
    ? [A] :
      ( relation(A)
      & relation_empty_yielding(A) ) ).

fof(reflexivity_r1_tarski,axiom,
    ! [A,B] : subset(A,A) ).

fof(t117_funct_1,axiom,
    ! [A,B] :
      ( ( relation(B)
        & function(B) )
     => ( in(A,relation_dom(B))
       => relation_image(B,singleton(A)) = singleton(apply(B,A)) ) ) ).

fof(t124_funct_1,conjecture,
    ! [A] :
      ( ( relation(A)
        & function(A) )
     => ( ! [B,C] : relation_image(A,set_difference(B,C)) = set_difference(relation_image(A,B),relation_image(A,C))
       => one_to_one(A) ) ) ).

fof(t1_subset,axiom,
    ! [A,B] :
      ( in(A,B)
     => element(A,B) ) ).

fof(t20_zfmisc_1,axiom,
    ! [A,B] :
      ( set_difference(singleton(A),singleton(B)) = singleton(A)
    <=> A != B ) ).

fof(t2_subset,axiom,
    ! [A,B] :
      ( element(A,B)
     => ( empty(B)
        | in(A,B) ) ) ).

fof(t37_xboole_1,axiom,
    ! [A,B] :
      ( set_difference(A,B) = empty_set
    <=> subset(A,B) ) ).

fof(t3_boole,axiom,
    ! [A] : set_difference(A,empty_set) = A ).

fof(t3_subset,axiom,
    ! [A,B] :
      ( element(A,powerset(B))
    <=> subset(A,B) ) ).

fof(t4_boole,axiom,
    ! [A] : set_difference(empty_set,A) = empty_set ).

fof(t4_subset,axiom,
    ! [A,B,C] :
      ( ( in(A,B)
        & element(B,powerset(C)) )
     => element(A,C) ) ).

fof(t5_subset,axiom,
    ! [A,B,C] :
      ~ ( in(A,B)
        & element(B,powerset(C))
        & empty(C) ) ).

fof(t6_boole,axiom,
    ! [A] :
      ( empty(A)
     => A = empty_set ) ).

fof(t7_boole,axiom,
    ! [A,B] :
      ~ ( in(A,B)
        & empty(B) ) ).

fof(t8_boole,axiom,
    ! [A,B] :
      ~ ( empty(A)
        & A != B
        & empty(B) ) ).

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