TPTP Problem File: SET648+3.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SET648+3 : TPTP v8.2.0. Released v2.2.0.
% Domain : Set Theory (Relations)
% Problem : Range of R (X to Y) a subset of Y => R is (domain of R to Y)
% Version : [Wor90] axioms : Reduced > Incomplete.
% English : If the range of a relation R from X to Y is a subset of Y, R is
% a relation from the domain of a relation R from X to Y and Y.
% Refs : [ILF] The ILF Group (1998), The ILF System: A Tool for the Int
% : [Wor90] Woronowicz (1990), Relations Defined on Sets
% Source : [ILF]
% Names : RELSET_1 (10) [Wor90]
% Status : Theorem
% Rating : 0.39 v8.1.0, 0.42 v7.5.0, 0.41 v7.4.0, 0.33 v7.3.0, 0.41 v7.2.0, 0.38 v7.1.0, 0.30 v7.0.0, 0.40 v6.4.0, 0.38 v6.3.0, 0.42 v6.2.0, 0.48 v6.1.0, 0.63 v6.0.0, 0.61 v5.5.0, 0.59 v5.4.0, 0.57 v5.3.0, 0.59 v5.2.0, 0.45 v5.1.0, 0.38 v4.1.0, 0.48 v4.0.0, 0.42 v3.7.0, 0.45 v3.5.0, 0.42 v3.4.0, 0.47 v3.3.0, 0.43 v3.2.0, 0.55 v3.1.0, 0.56 v2.7.0, 0.50 v2.6.0, 0.43 v2.5.0, 0.50 v2.4.0, 0.75 v2.3.0, 0.67 v2.2.1
% Syntax : Number of formulae : 27 ( 2 unt; 0 def)
% Number of atoms : 102 ( 1 equ)
% Maximal formula atoms : 7 ( 3 avg)
% Number of connectives : 79 ( 4 ~; 0 |; 12 &)
% ( 9 <=>; 54 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 6 ( 5 usr; 0 prp; 1-2 aty)
% Number of functors : 10 ( 10 usr; 2 con; 0-2 aty)
% Number of variables : 57 ( 49 !; 8 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments :
%------------------------------------------------------------------------------
%---- line(boole - th(29),1909428)
fof(p1,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,set_type)
=> ( ( subset(B,C)
& subset(C,D) )
=> subset(B,D) ) ) ) ) ).
%---- line(relat_1 - th(21),1917997)
fof(p2,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> subset(B,cross_product(domain_of(B),range_of(B))) ) ).
%---- line(zfmisc_1 - th(118),1905443)
fof(p3,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,set_type)
=> ( subset(B,C)
=> ( subset(cross_product(B,D),cross_product(C,D))
& subset(cross_product(D,B),cross_product(D,C)) ) ) ) ) ) ).
%---- line(relset_1 - df(1),1916080)
fof(p4,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( ! [D] :
( ilf_type(D,subset_type(cross_product(B,C)))
=> ilf_type(D,relation_type(B,C)) )
& ! [E] :
( ilf_type(E,relation_type(B,C))
=> ilf_type(E,subset_type(cross_product(B,C))) ) ) ) ) ).
%---- type_nonempty(line(relset_1 - df(1),1916080))
fof(p5,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ? [D] : ilf_type(D,relation_type(C,B)) ) ) ).
%---- line(relat_1 - df(4),1917872)
fof(p6,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( member(C,domain_of(B))
<=> ? [D] :
( ilf_type(D,set_type)
& member(ordered_pair(C,D),B) ) ) ) ) ).
%---- declaration(line(relat_1 - df(4),1917872))
fof(p7,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ilf_type(domain_of(B),set_type) ) ).
%---- line(relat_1 - df(5),1917958)
fof(p8,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( member(C,range_of(B))
<=> ? [D] :
( ilf_type(D,set_type)
& member(ordered_pair(D,C),B) ) ) ) ) ).
%---- declaration(line(relat_1 - df(5),1917958))
fof(p9,axiom,
! [B] :
( ilf_type(B,binary_relation_type)
=> ilf_type(range_of(B),set_type) ) ).
%---- line(relat_1 - axiom70,1917641)
fof(p10,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( ilf_type(B,binary_relation_type)
<=> ( relation_like(B)
& ilf_type(B,set_type) ) ) ) ).
%---- type_nonempty(line(relat_1 - axiom70,1917641))
fof(p11,axiom,
? [B] : ilf_type(B,binary_relation_type) ).
%---- line(tarski - df(3),1832749)
fof(p12,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( subset(B,C)
<=> ! [D] :
( ilf_type(D,set_type)
=> ( member(D,B)
=> member(D,C) ) ) ) ) ) ).
%---- declaration(op(cross_product,2,function))
fof(p13,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(cross_product(B,C),set_type) ) ) ).
%---- declaration(op(ordered_pair,2,function))
fof(p14,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ilf_type(ordered_pair(B,C),set_type) ) ) ).
%---- line(hidden - axiom71,1832648)
fof(p15,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( ilf_type(C,subset_type(B))
<=> ilf_type(C,member_type(power_set(B))) ) ) ) ).
%---- type_nonempty(line(hidden - axiom71,1832648))
fof(p16,axiom,
! [B] :
( ilf_type(B,set_type)
=> ? [C] : ilf_type(C,subset_type(B)) ) ).
%---- property(reflexivity,op(subset,2,predicate))
fof(p17,axiom,
! [B] :
( ilf_type(B,set_type)
=> subset(B,B) ) ).
%---- line(relat_1 - df(1),1917627)
fof(p18,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( relation_like(B)
<=> ! [C] :
( ilf_type(C,set_type)
=> ( member(C,B)
=> ? [D] :
( ilf_type(D,set_type)
& ? [E] :
( ilf_type(E,set_type)
& C = ordered_pair(D,E) ) ) ) ) ) ) ).
%---- conditional_cluster(axiom74,relation_like)
fof(p19,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ! [D] :
( ilf_type(D,subset_type(cross_product(B,C)))
=> relation_like(D) ) ) ) ).
%---- line(hidden - axiom75,1832644)
fof(p20,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,set_type)
=> ( member(B,power_set(C))
<=> ! [D] :
( ilf_type(D,set_type)
=> ( member(D,B)
=> member(D,C) ) ) ) ) ) ).
%---- declaration(line(hidden - axiom75,1832644))
fof(p21,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( ~ empty(power_set(B))
& ilf_type(power_set(B),set_type) ) ) ).
%---- line(hidden - axiom76,1832640)
fof(p22,axiom,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ( ~ empty(C)
& ilf_type(C,set_type) )
=> ( ilf_type(B,member_type(C))
<=> member(B,C) ) ) ) ).
%---- type_nonempty(line(hidden - axiom76,1832640))
fof(p23,axiom,
! [B] :
( ( ~ empty(B)
& ilf_type(B,set_type) )
=> ? [C] : ilf_type(C,member_type(B)) ) ).
%---- line(hidden - axiom77,1832628)
fof(p24,axiom,
! [B] :
( ilf_type(B,set_type)
=> ( empty(B)
<=> ! [C] :
( ilf_type(C,set_type)
=> ~ member(C,B) ) ) ) ).
%---- conditional_cluster(axiom78,empty)
fof(p25,axiom,
! [B] :
( ( empty(B)
& ilf_type(B,set_type) )
=> relation_like(B) ) ).
%---- declaration(set)
fof(p26,axiom,
! [B] : ilf_type(B,set_type) ).
%---- line(relset_1 - th(10),1916164)
fof(prove_relset_1_10,conjecture,
! [B] :
( ilf_type(B,set_type)
=> ! [C] :
( ilf_type(C,binary_relation_type)
=> ( subset(range_of(C),B)
=> ilf_type(C,relation_type(domain_of(C),B)) ) ) ) ).
%------------------------------------------------------------------------------