TPTP Problem File: CAT034+2.p
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%------------------------------------------------------------------------------
% File : CAT034+2 : TPTP v8.2.0. Released v3.4.0.
% Domain : Category Theory
% Problem : Yoneda Embedding T04
% Version : [Urb08] axioms : Especial.
% English :
% Refs : [Woj97] Wojciechowski (1997), Yoneda Embedding
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : t4_yoneda_1 [Urb08]
% Status : Theorem
% Rating : 0.94 v7.5.0, 0.97 v7.1.0, 0.96 v7.0.0, 1.00 v3.4.0
% Syntax : Number of formulae : 3806 (1054 unt; 0 def)
% Number of atoms : 17044 (3322 equ)
% Maximal formula atoms : 52 ( 4 avg)
% Number of connectives : 14636 (1398 ~; 191 |;6091 &)
% ( 595 <=>;6361 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 6 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 197 ( 195 usr; 1 prp; 0-6 aty)
% Number of functors : 794 ( 794 usr; 294 con; 0-9 aty)
% Number of variables : 9811 (9313 !; 498 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Bushy version: includes all articles that contribute axioms to the
% Normal version.
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : The problem encoding is based on set theory.
%------------------------------------------------------------------------------
include('Axioms/SET007/SET007+0.ax').
include('Axioms/SET007/SET007+1.ax').
include('Axioms/SET007/SET007+2.ax').
include('Axioms/SET007/SET007+3.ax').
include('Axioms/SET007/SET007+5.ax').
include('Axioms/SET007/SET007+6.ax').
include('Axioms/SET007/SET007+7.ax').
include('Axioms/SET007/SET007+9.ax').
include('Axioms/SET007/SET007+10.ax').
include('Axioms/SET007/SET007+11.ax').
include('Axioms/SET007/SET007+14.ax').
include('Axioms/SET007/SET007+16.ax').
include('Axioms/SET007/SET007+17.ax').
include('Axioms/SET007/SET007+18.ax').
include('Axioms/SET007/SET007+20.ax').
include('Axioms/SET007/SET007+21.ax').
include('Axioms/SET007/SET007+22.ax').
include('Axioms/SET007/SET007+25.ax').
include('Axioms/SET007/SET007+26.ax').
include('Axioms/SET007/SET007+31.ax').
include('Axioms/SET007/SET007+34.ax').
include('Axioms/SET007/SET007+35.ax').
include('Axioms/SET007/SET007+54.ax').
include('Axioms/SET007/SET007+55.ax').
include('Axioms/SET007/SET007+61.ax').
include('Axioms/SET007/SET007+76.ax').
include('Axioms/SET007/SET007+77.ax').
include('Axioms/SET007/SET007+79.ax').
include('Axioms/SET007/SET007+202.ax').
include('Axioms/SET007/SET007+237.ax').
include('Axioms/SET007/SET007+293.ax').
include('Axioms/SET007/SET007+299.ax').
include('Axioms/SET007/SET007+308.ax').
include('Axioms/SET007/SET007+322.ax').
%------------------------------------------------------------------------------
fof(dt_k1_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ( v2_cat_1(k1_yoneda_1(A))
& l1_cat_1(k1_yoneda_1(A)) ) ) ).
fof(dt_k2_yoneda_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> m2_cat_1(k2_yoneda_1(A,B),A,k1_yoneda_1(A)) ) ).
fof(dt_k3_yoneda_1,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u2_cat_1(A)) )
=> m2_nattra_1(k3_yoneda_1(A,B),A,k1_yoneda_1(A),k2_yoneda_1(A,k3_cat_1(A,B)),k2_yoneda_1(A,k2_cat_1(A,B))) ) ).
fof(dt_k4_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> m1_oppcat_1(k4_yoneda_1(A),A,k12_nattra_1(A,k1_yoneda_1(A))) ) ).
fof(dt_k5_yoneda_1,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m1_oppcat_1(C,A,B)
& m1_subset_1(D,u1_cat_1(A)) )
=> m1_subset_1(k5_yoneda_1(A,B,C,D),u1_cat_1(B)) ) ).
fof(d1_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> k1_yoneda_1(A) = k12_ens_1(k17_ens_1(A)) ) ).
fof(t1_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( m1_subset_1(D,u2_cat_1(k1_yoneda_1(A)))
=> ! [E] :
( m1_subset_1(E,u2_cat_1(k1_yoneda_1(A)))
=> ( ( k3_cat_1(k1_yoneda_1(A),D) = k2_cat_1(k1_yoneda_1(A),E)
& k4_tarski(k12_cat_2(k1_yoneda_1(A),k1_yoneda_1(A),k2_cat_1(k1_yoneda_1(A),D),k3_cat_1(k1_yoneda_1(A),D)),B) = D
& k4_tarski(k12_cat_2(k1_yoneda_1(A),k1_yoneda_1(A),k2_cat_1(k1_yoneda_1(A),E),k3_cat_1(k1_yoneda_1(A),E)),C) = E )
=> k4_tarski(k12_cat_2(k1_yoneda_1(A),k1_yoneda_1(A),k2_cat_1(k1_yoneda_1(A),D),k3_cat_1(k1_yoneda_1(A),E)),k5_relat_1(B,C)) = k4_cat_1(k1_yoneda_1(A),D,E) ) ) ) ) ) ) ).
fof(t2_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> m2_cat_1(k20_ens_1(A,B),A,k1_yoneda_1(A)) ) ) ).
fof(d2_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> k2_yoneda_1(A,B) = k20_ens_1(A,B) ) ) ).
fof(t3_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> r2_nattra_1(A,k1_yoneda_1(A),k2_yoneda_1(A,k3_cat_1(A,B)),k2_yoneda_1(A,k2_cat_1(A,B))) ) ) ).
fof(d3_yoneda_1,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> ! [C] :
( m2_nattra_1(C,A,k1_yoneda_1(A),k2_yoneda_1(A,k3_cat_1(A,B)),k2_yoneda_1(A,k2_cat_1(A,B)))
=> ( C = k3_yoneda_1(A,B)
<=> ! [D] :
( m1_subset_1(D,u1_cat_1(A))
=> k5_nattra_1(A,k1_yoneda_1(A),k2_yoneda_1(A,k3_cat_1(A,B)),k2_yoneda_1(A,k2_cat_1(A,B)),C,D) = k4_tarski(k4_tarski(k6_cat_1(A,k3_cat_1(A,B),D),k6_cat_1(A,k2_cat_1(A,B),D)),k22_ens_1(A,B,k10_cat_1(A,D))) ) ) ) ) ) ).
fof(t4_yoneda_1,conjecture,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u2_cat_1(A))
=> m1_subset_1(k4_tarski(k4_tarski(k2_yoneda_1(A,k3_cat_1(A,B)),k2_yoneda_1(A,k2_cat_1(A,B))),k3_yoneda_1(A,B)),u2_cat_1(k12_nattra_1(A,k1_yoneda_1(A)))) ) ) ).
%------------------------------------------------------------------------------