## SET007 Axioms: SET007+514.ax

```%------------------------------------------------------------------------------
% File     : SET007+514 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Yoneda Embedding
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : yoneda_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   21 (   0 unit)
%            Number of atoms       :  120 (  17 equality)
%            Maximal formula depth :   18 (   8 average)
%            Number of connectives :  102 (   3 ~  ;   0  |;  39  &)
%                                         (   4 <=>;  56 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   15 (   0 propositional; 1-5 arity)
%            Number of functors    :   24 (   1 constant; 0-6 arity)
%            Number of variables   :   56 (   0 singleton;  55 !;   1 ?)
%            Maximal term depth    :    5 (   2 average)
% SPC      :

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
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,axiom,(
! [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)))) ) ) )).

fof(d4_yoneda_1,axiom,(
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_oppcat_1(B,A,k12_nattra_1(A,k1_yoneda_1(A)))
=> ( B = k4_yoneda_1(A)
<=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> k8_funct_2(u2_cat_1(A),u2_cat_1(k12_nattra_1(A,k1_yoneda_1(A))),B,C) = k4_tarski(k4_tarski(k2_yoneda_1(A,k3_cat_1(A,C)),k2_yoneda_1(A,k2_cat_1(A,C))),k3_yoneda_1(A,C)) ) ) ) ) )).

fof(d5_yoneda_1,axiom,(
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_oppcat_1(C,A,B)
=> ! [D] :
( m1_subset_1(D,u1_cat_1(A))
=> k5_yoneda_1(A,B,C,D) = k8_funct_2(u1_cat_1(A),u1_cat_1(B),k12_cat_1(A,B,C),D) ) ) ) ) )).

fof(t5_yoneda_1,axiom,(
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m2_cat_1(B,A,k12_nattra_1(A,k1_yoneda_1(A)))
=> ( ( v2_funct_1(k12_cat_1(A,k12_nattra_1(A,k1_yoneda_1(A)),B))
& v10_cat_1(B,A,k12_nattra_1(A,k1_yoneda_1(A))) )
=> v2_funct_1(B) ) ) ) )).

fof(d6_yoneda_1,axiom,(
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_oppcat_1(C,A,B)
=> ( v1_yoneda_1(C,A,B)
<=> ! [D] :
( m1_subset_1(D,u1_cat_1(A))
=> ! [E] :
( m1_subset_1(E,u1_cat_1(A))
=> ( k6_cat_1(A,D,E) != k1_xboole_0
=> ! [F] :
( m1_cat_1(F,A,D,E)
=> ! [G] :
( m1_cat_1(G,A,D,E)
=> ( k8_funct_2(u2_cat_1(A),u2_cat_1(B),C,F) = k8_funct_2(u2_cat_1(A),u2_cat_1(B),C,G)
=> F = G ) ) ) ) ) ) ) ) ) ) )).

fof(t6_yoneda_1,axiom,(
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_oppcat_1(B,A,k12_nattra_1(A,k1_yoneda_1(A)))
=> ( ( v2_funct_1(k12_cat_1(A,k12_nattra_1(A,k1_yoneda_1(A)),B))
& v1_yoneda_1(B,A,k12_nattra_1(A,k1_yoneda_1(A))) )
=> v2_funct_1(B) ) ) ) )).

fof(t7_yoneda_1,axiom,(
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> v1_yoneda_1(k4_yoneda_1(A),A,k12_nattra_1(A,k1_yoneda_1(A))) ) )).

fof(t8_yoneda_1,axiom,(
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> v2_funct_1(k4_yoneda_1(A)) ) )).

fof(d7_yoneda_1,axiom,(
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_oppcat_1(C,A,B)
=> ( v2_yoneda_1(C,A,B)
<=> ! [D] :
( m1_subset_1(D,u1_cat_1(A))
=> ! [E] :
( m1_subset_1(E,u1_cat_1(A))
=> ( k6_cat_1(B,k5_yoneda_1(A,B,C,E),k5_yoneda_1(A,B,C,D)) != k1_xboole_0
=> ! [F] :
( m1_cat_1(F,B,k5_yoneda_1(A,B,C,E),k5_yoneda_1(A,B,C,D))
=> ( k6_cat_1(A,D,E) != k1_xboole_0
& ? [G] :
( m1_cat_1(G,A,D,E)
& F = k8_funct_2(u2_cat_1(A),u2_cat_1(B),C,G) ) ) ) ) ) ) ) ) ) ) )).

fof(t9_yoneda_1,axiom,(
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> v2_yoneda_1(k4_yoneda_1(A),A,k12_nattra_1(A,k1_yoneda_1(A))) ) )).

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)) ) )).
%------------------------------------------------------------------------------
```