Metamath Proof Explorer

Theorem fuconuniqex

Description: The object function of functor composition does not in general determine its morphism function, even when both source categories are the trivial category. The witness category is the explicit two-object, two-parallel-arrow category of par2cat . (Contributed by Zhi Wang, 14-Sep-2026)

Ref Expression
Assertion fuconuniqex 
|- E. e e. Cat E. z ( ( 1st ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F e ) ) ( ( ( ( SetCat ` 1o ) FuncCat e ) Xc. ( ( SetCat ` 1o ) FuncCat ( SetCat ` 1o ) ) ) Func ( ( SetCat ` 1o ) FuncCat e ) ) z /\ z =/= ( 2nd ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F e ) ) )

Proof

Step Hyp Ref Expression
1 par2cat 
 |-  { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } e. Cat
2 eqid 
 |-  ( SetCat ` 1o ) = ( SetCat ` 1o )
3 eqid 
 |-  { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } = { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. }
4 eqid 
 |-  ( ( SetCat ` 1o ) FuncCat { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) = ( ( SetCat ` 1o ) FuncCat { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } )
5 eqid 
 |-  ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) = ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } )
6 eqid 
 |-  ( ( ( SetCat ` 1o ) FuncCat { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) Xc. ( ( SetCat ` 1o ) FuncCat ( SetCat ` 1o ) ) ) = ( ( ( SetCat ` 1o ) FuncCat { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) Xc. ( ( SetCat ` 1o ) FuncCat ( SetCat ` 1o ) ) )
7 2 3 4 5 6 fuconuniq2 
 |-  E. z ( ( 1st ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) ) ( ( ( ( SetCat ` 1o ) FuncCat { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) Xc. ( ( SetCat ` 1o ) FuncCat ( SetCat ` 1o ) ) ) Func ( ( SetCat ` 1o ) FuncCat { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) ) z /\ z =/= ( 2nd ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) ) )
8 oveq2 
 |-  ( e = { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } -> ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F e ) = ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) )
9 8 fveq2d 
 |-  ( e = { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } -> ( 1st ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F e ) ) = ( 1st ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) ) )
10 oveq2 
 |-  ( e = { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } -> ( ( SetCat ` 1o ) FuncCat e ) = ( ( SetCat ` 1o ) FuncCat { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) )
11 10 oveq1d 
 |-  ( e = { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } -> ( ( ( SetCat ` 1o ) FuncCat e ) Xc. ( ( SetCat ` 1o ) FuncCat ( SetCat ` 1o ) ) ) = ( ( ( SetCat ` 1o ) FuncCat { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) Xc. ( ( SetCat ` 1o ) FuncCat ( SetCat ` 1o ) ) ) )
12 11 10 oveq12d 
 |-  ( e = { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } -> ( ( ( ( SetCat ` 1o ) FuncCat e ) Xc. ( ( SetCat ` 1o ) FuncCat ( SetCat ` 1o ) ) ) Func ( ( SetCat ` 1o ) FuncCat e ) ) = ( ( ( ( SetCat ` 1o ) FuncCat { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) Xc. ( ( SetCat ` 1o ) FuncCat ( SetCat ` 1o ) ) ) Func ( ( SetCat ` 1o ) FuncCat { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) ) )
13 eqidd 
 |-  ( e = { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } -> z = z )
14 9 12 13 breq123d 
 |-  ( e = { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } -> ( ( 1st ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F e ) ) ( ( ( ( SetCat ` 1o ) FuncCat e ) Xc. ( ( SetCat ` 1o ) FuncCat ( SetCat ` 1o ) ) ) Func ( ( SetCat ` 1o ) FuncCat e ) ) z <-> ( 1st ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) ) ( ( ( ( SetCat ` 1o ) FuncCat { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) Xc. ( ( SetCat ` 1o ) FuncCat ( SetCat ` 1o ) ) ) Func ( ( SetCat ` 1o ) FuncCat { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) ) z ) )
15 8 fveq2d 
 |-  ( e = { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } -> ( 2nd ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F e ) ) = ( 2nd ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) ) )
16 15 neeq2d 
 |-  ( e = { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } -> ( z =/= ( 2nd ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F e ) ) <-> z =/= ( 2nd ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) ) ) )
17 14 16 anbi12d 
 |-  ( e = { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } -> ( ( ( 1st ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F e ) ) ( ( ( ( SetCat ` 1o ) FuncCat e ) Xc. ( ( SetCat ` 1o ) FuncCat ( SetCat ` 1o ) ) ) Func ( ( SetCat ` 1o ) FuncCat e ) ) z /\ z =/= ( 2nd ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F e ) ) ) <-> ( ( 1st ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) ) ( ( ( ( SetCat ` 1o ) FuncCat { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) Xc. ( ( SetCat ` 1o ) FuncCat ( SetCat ` 1o ) ) ) Func ( ( SetCat ` 1o ) FuncCat { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) ) z /\ z =/= ( 2nd ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) ) ) ) )
18 17 exbidv 
 |-  ( e = { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } -> ( E. z ( ( 1st ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F e ) ) ( ( ( ( SetCat ` 1o ) FuncCat e ) Xc. ( ( SetCat ` 1o ) FuncCat ( SetCat ` 1o ) ) ) Func ( ( SetCat ` 1o ) FuncCat e ) ) z /\ z =/= ( 2nd ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F e ) ) ) <-> E. z ( ( 1st ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) ) ( ( ( ( SetCat ` 1o ) FuncCat { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) Xc. ( ( SetCat ` 1o ) FuncCat ( SetCat ` 1o ) ) ) Func ( ( SetCat ` 1o ) FuncCat { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) ) z /\ z =/= ( 2nd ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) ) ) ) )
19 18 rspcev 
 |-  ( ( { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } e. Cat /\ E. z ( ( 1st ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) ) ( ( ( ( SetCat ` 1o ) FuncCat { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) Xc. ( ( SetCat ` 1o ) FuncCat ( SetCat ` 1o ) ) ) Func ( ( SetCat ` 1o ) FuncCat { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) ) z /\ z =/= ( 2nd ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F { <. ( Base ` ndx ) , 2o >. , <. ( Hom ` ndx ) , ( m e. 2o , n e. 2o |-> if ( m = n , { 2o } , if ( ( m = (/) /\ n = 1o ) , 2o , (/) ) ) ) >. , <. ( comp ` ndx ) , ( p e. ( 2o X. 2o ) , q e. 2o |-> ( r e. 3o , s e. 3o |-> ( r i^i s ) ) ) >. } ) ) ) ) -> E. e e. Cat E. z ( ( 1st ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F e ) ) ( ( ( ( SetCat ` 1o ) FuncCat e ) Xc. ( ( SetCat ` 1o ) FuncCat ( SetCat ` 1o ) ) ) Func ( ( SetCat ` 1o ) FuncCat e ) ) z /\ z =/= ( 2nd ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F e ) ) ) )
20 1 7 19 mp2an 
 |-  E. e e. Cat E. z ( ( 1st ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F e ) ) ( ( ( ( SetCat ` 1o ) FuncCat e ) Xc. ( ( SetCat ` 1o ) FuncCat ( SetCat ` 1o ) ) ) Func ( ( SetCat ` 1o ) FuncCat e ) ) z /\ z =/= ( 2nd ` ( <. ( SetCat ` 1o ) , ( SetCat ` 1o ) >. o.F e ) ) )