TPTP Problem File: SEU078+1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SEU078+1 : TPTP v8.2.0. Released v3.2.0.
% Domain : Set theory
% Problem : Functions and their basic properties, theorem 159
% 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__t159_funct_1 [Urb06]
% Status : Theorem
% Rating : 0.69 v8.1.0, 0.64 v7.5.0, 0.72 v7.4.0, 0.67 v7.3.0, 0.69 v7.2.0, 0.66 v7.1.0, 0.61 v7.0.0, 0.67 v6.4.0, 0.65 v6.3.0, 0.62 v6.2.0, 0.76 v6.1.0, 0.83 v6.0.0, 0.78 v5.5.0, 0.81 v5.4.0, 0.82 v5.3.0, 0.85 v5.2.0, 0.70 v5.1.0, 0.76 v5.0.0, 0.79 v4.1.0, 0.78 v4.0.1, 0.74 v4.0.0, 0.75 v3.5.0, 0.68 v3.4.0, 0.79 v3.3.0, 0.86 v3.2.0
% Syntax : Number of formulae : 43 ( 8 unt; 0 def)
% Number of atoms : 110 ( 11 equ)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 85 ( 18 ~; 2 |; 34 &)
% ( 11 <=>; 20 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 10 ( 9 usr; 0 prp; 1-2 aty)
% Number of functors : 7 ( 7 usr; 1 con; 0-2 aty)
% Number of variables : 70 ( 57 !; 13 ?)
% 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(d13_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ! [B,C] :
( C = relation_inverse_image(A,B)
<=> ! [D] :
( in(D,C)
<=> ( in(D,relation_dom(A))
& in(apply(A,D),B) ) ) ) ) ).
fof(d1_tarski,axiom,
! [A,B] :
( B = singleton(A)
<=> ! [C] :
( in(C,B)
<=> C = A ) ) ).
fof(d5_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ! [B] :
( B = relation_rng(A)
<=> ! [C] :
( in(C,B)
<=> ? [D] :
( in(D,relation_dom(A))
& C = apply(A,D) ) ) ) ) ).
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(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(fc6_relat_1,axiom,
! [A] :
( ( ~ empty(A)
& relation(A) )
=> ~ empty(relation_rng(A)) ) ).
fof(fc7_relat_1,axiom,
! [A] :
( empty(A)
=> ( empty(relation_dom(A))
& relation(relation_dom(A)) ) ) ).
fof(fc8_relat_1,axiom,
! [A] :
( empty(A)
=> ( empty(relation_rng(A))
& relation(relation_rng(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(symmetry_r1_xboole_0,axiom,
! [A,B] :
( disjoint(A,B)
=> disjoint(B,A) ) ).
fof(t144_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ( ! [B] :
~ ( in(B,relation_rng(A))
& ! [C] : relation_inverse_image(A,singleton(B)) != singleton(C) )
<=> one_to_one(A) ) ) ).
fof(t159_funct_1,conjecture,
! [A] :
( ( relation(A)
& function(A) )
=> ( one_to_one(A)
<=> ! [B] :
? [C] : subset(relation_inverse_image(A,singleton(B)),singleton(C)) ) ) ).
fof(t173_relat_1,axiom,
! [A,B] :
( relation(B)
=> ( relation_inverse_image(B,A) = empty_set
<=> disjoint(relation_rng(B),A) ) ) ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t2_xboole_1,axiom,
! [A] : subset(empty_set,A) ).
fof(t39_zfmisc_1,axiom,
! [A,B] :
( subset(A,singleton(B))
<=> ( A = empty_set
| A = singleton(B) ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( element(A,powerset(B))
<=> subset(A,B) ) ).
fof(t4_subset,axiom,
! [A,B,C] :
( ( in(A,B)
& element(B,powerset(C)) )
=> element(A,C) ) ).
fof(t56_zfmisc_1,axiom,
! [A,B] :
( ~ in(A,B)
=> disjoint(singleton(A),B) ) ).
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) ) ).
%------------------------------------------------------------------------------