TPTP Problem File: SEU004+1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SEU004+1 : TPTP v9.0.0. Released v3.2.0.
% Domain : Set theory
% Problem : Functions and their basic properties, theorem 33
% 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__t33_funct_1 [Urb06]
% Status : Theorem
% Rating : 1.00 v3.7.0, 0.95 v3.5.0, 1.00 v3.2.0
% Syntax : Number of formulae : 44 ( 8 unt; 0 def)
% Number of atoms : 151 ( 35 equ)
% Maximal formula atoms : 23 ( 3 avg)
% Number of connectives : 128 ( 21 ~; 1 |; 56 &)
% ( 5 <=>; 45 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 8 ( 7 usr; 0 prp; 1-2 aty)
% Number of functors : 8 ( 8 usr; 3 con; 0-2 aty)
% Number of variables : 81 ( 69 !; 12 ?)
% 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(d10_xboole_0,axiom,
! [A,B] :
( A = B
<=> ( subset(A,B)
& subset(B,A) ) ) ).
fof(d3_tarski,axiom,
! [A,B] :
( subset(A,B)
<=> ! [C] :
( in(C,A)
=> in(C,B) ) ) ).
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(dt_k5_relat_1,axiom,
! [A,B] :
( ( relation(A)
& relation(B) )
=> relation(relation_composition(A,B)) ) ).
fof(existence_m1_subset_1,axiom,
! [A] :
? [B] : element(B,A) ).
fof(fc10_relat_1,axiom,
! [A,B] :
( ( empty(A)
& relation(B) )
=> ( empty(relation_composition(B,A))
& relation(relation_composition(B,A)) ) ) ).
fof(fc12_relat_1,axiom,
( empty(empty_set)
& relation(empty_set)
& relation_empty_yielding(empty_set) ) ).
fof(fc1_funct_1,axiom,
! [A,B] :
( ( relation(A)
& function(A)
& relation(B)
& function(B) )
=> ( relation(relation_composition(A,B))
& function(relation_composition(A,B)) ) ) ).
fof(fc1_subset_1,axiom,
! [A] : ~ empty(powerset(A)) ).
fof(fc1_xboole_0,axiom,
empty(empty_set) ).
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(fc9_relat_1,axiom,
! [A,B] :
( ( empty(A)
& relation(B) )
=> ( empty(relation_composition(A,B))
& relation(relation_composition(A,B)) ) ) ).
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_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_relat_1,axiom,
? [A] :
( relation(A)
& relation_empty_yielding(A) ) ).
fof(reflexivity_r1_tarski,axiom,
! [A,B] : subset(A,A) ).
fof(s2_funct_1__e8_26_1__funct_1,axiom,
! [A,B] :
( ( ! [C,D,E] :
( ( in(C,A)
& ( C = B
=> D = n1 )
& ( C != B
=> D = n0 )
& ( C = B
=> E = n1 )
& ( C != B
=> E = n0 ) )
=> D = E )
& ! [C] :
~ ( in(C,A)
& ! [D] :
~ ( ( C = B
=> D = n1 )
& ( C != B
=> D = n0 ) ) ) )
=> ? [C] :
( relation(C)
& function(C)
& relation_dom(C) = A
& ! [D] :
( in(D,A)
=> ( ( D = B
=> apply(C,D) = n1 )
& ( D != B
=> apply(C,D) = n0 ) ) ) ) ) ).
fof(s3_funct_1__e3_26_1__funct_1,axiom,
! [A] :
? [B] :
( relation(B)
& function(B)
& relation_dom(B) = A
& ! [C] :
( in(C,A)
=> apply(B,C) = n0 ) ) ).
fof(spc0_boole,axiom,
empty(n0) ).
fof(spc1_boole,axiom,
~ empty(n1) ).
fof(t1_subset,axiom,
! [A,B] :
( in(A,B)
=> element(A,B) ) ).
fof(t22_funct_1,axiom,
! [A,B] :
( ( relation(B)
& function(B) )
=> ! [C] :
( ( relation(C)
& function(C) )
=> ( in(A,relation_dom(relation_composition(C,B)))
=> apply(relation_composition(C,B),A) = apply(B,apply(C,A)) ) ) ) ).
fof(t2_subset,axiom,
! [A,B] :
( element(A,B)
=> ( empty(B)
| in(A,B) ) ) ).
fof(t33_funct_1,conjecture,
! [A,B] :
( ( relation(B)
& function(B) )
=> ( ( subset(relation_rng(B),A)
& ! [C] :
( ( relation(C)
& function(C) )
=> ! [D] :
( ( relation(D)
& function(D) )
=> ( ( relation_dom(C) = A
& relation_dom(D) = A
& relation_composition(B,C) = relation_composition(B,D) )
=> C = D ) ) ) )
=> A = relation_rng(B) ) ) ).
fof(t3_subset,axiom,
! [A,B] :
( element(A,powerset(B))
<=> subset(A,B) ) ).
fof(t46_relat_1,axiom,
! [A] :
( relation(A)
=> ! [B] :
( relation(B)
=> ( subset(relation_rng(A),relation_dom(B))
=> relation_dom(relation_composition(A,B)) = relation_dom(A) ) ) ) ).
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) ) ).
fof(t9_funct_1,axiom,
! [A] :
( ( relation(A)
& function(A) )
=> ! [B] :
( ( relation(B)
& function(B) )
=> ( ( relation_dom(A) = relation_dom(B)
& ! [C] :
( in(C,relation_dom(A))
=> apply(A,C) = apply(B,C) ) )
=> A = B ) ) ) ).
%------------------------------------------------------------------------------