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Theorem yonedalem3b 15085
Description: Lemma for yoneda 15089. (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
yoneda.y  |-  Y  =  (Yon `  C )
yoneda.b  |-  B  =  ( Base `  C
)
yoneda.1  |-  .1.  =  ( Id `  C )
yoneda.o  |-  O  =  (oppCat `  C )
yoneda.s  |-  S  =  ( SetCat `  U )
yoneda.t  |-  T  =  ( SetCat `  V )
yoneda.q  |-  Q  =  ( O FuncCat  S )
yoneda.h  |-  H  =  (HomF
`  Q )
yoneda.r  |-  R  =  ( ( Q  X.c  O
) FuncCat  T )
yoneda.e  |-  E  =  ( O evalF  S )
yoneda.z  |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )
yoneda.c  |-  ( ph  ->  C  e.  Cat )
yoneda.w  |-  ( ph  ->  V  e.  W )
yoneda.u  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
yoneda.v  |-  ( ph  ->  ( ran  ( Hom f  `  Q )  u.  U
)  C_  V )
yonedalem21.f  |-  ( ph  ->  F  e.  ( O 
Func  S ) )
yonedalem21.x  |-  ( ph  ->  X  e.  B )
yonedalem22.g  |-  ( ph  ->  G  e.  ( O 
Func  S ) )
yonedalem22.p  |-  ( ph  ->  P  e.  B )
yonedalem22.a  |-  ( ph  ->  A  e.  ( F ( O Nat  S ) G ) )
yonedalem22.k  |-  ( ph  ->  K  e.  ( P ( Hom  `  C
) X ) )
yonedalem3.m  |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y
) `  x )
( O Nat  S ) f )  |->  ( ( a `  x ) `
 (  .1.  `  x ) ) ) )
Assertion
Ref Expression
yonedalem3b  |-  ( ph  ->  ( ( G M P ) ( <.
( F ( 1st `  Z ) X ) ,  ( G ( 1st `  Z ) P ) >. (comp `  T ) ( G ( 1st `  E
) P ) ) ( A ( <. F ,  X >. ( 2nd `  Z )
<. G ,  P >. ) K ) )  =  ( ( A (
<. F ,  X >. ( 2nd `  E )
<. G ,  P >. ) K ) ( <.
( F ( 1st `  Z ) X ) ,  ( F ( 1st `  E ) X ) >. (comp `  T ) ( G ( 1st `  E
) P ) ) ( F M X ) ) )
Distinct variable groups:    f, a, x,  .1.    A, a    C, a, f, x    E, a, f    F, a, f, x    K, a    B, a, f, x    G, a, f, x    O, a, f, x    S, a, f, x    Q, a, f, x    T, f    P, a, f, x    ph, a,
f, x    Y, a,
f, x    Z, a,
f, x    X, a,
f, x
Allowed substitution hints:    A( x, f)    R( x, f, a)    T( x, a)    U( x, f, a)    E( x)    H( x, f, a)    K( x, f)    M( x, f, a)    V( x, f, a)    W( x, f, a)

Proof of Theorem yonedalem3b
Dummy variables  b 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6098 . . . . . . . 8  |-  ( b  =  a  ->  ( A ( <. (
( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) b )  =  ( A ( <. ( ( 1st `  Y ) `  X
) ,  F >. (comp `  Q ) G ) a ) )
21oveq1d 6105 . . . . . . 7  |-  ( b  =  a  ->  (
( A ( <.
( ( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) b ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) )  =  ( ( A ( <. ( ( 1st `  Y ) `  X
) ,  F >. (comp `  Q ) G ) a ) ( <.
( ( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) )
32fveq1d 5690 . . . . . 6  |-  ( b  =  a  ->  (
( ( A (
<. ( ( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) b ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) `
 P )  =  ( ( ( A ( <. ( ( 1st `  Y ) `  X
) ,  F >. (comp `  Q ) G ) a ) ( <.
( ( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) `
 P ) )
43fveq1d 5690 . . . . 5  |-  ( b  =  a  ->  (
( ( ( A ( <. ( ( 1st `  Y ) `  X
) ,  F >. (comp `  Q ) G ) b ) ( <.
( ( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) `
 P ) `  (  .1.  `  P )
)  =  ( ( ( ( A (
<. ( ( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) a ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) `
 P ) `  (  .1.  `  P )
) )
54cbvmptv 4380 . . . 4  |-  ( b  e.  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F )  |->  ( ( ( ( A ( <.
( ( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) b ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) `
 P ) `  (  .1.  `  P )
) )  =  ( a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  |->  ( ( ( ( A (
<. ( ( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) a ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) `
 P ) `  (  .1.  `  P )
) )
6 yoneda.q . . . . . . . . 9  |-  Q  =  ( O FuncCat  S )
7 eqid 2441 . . . . . . . . 9  |-  ( O Nat 
S )  =  ( O Nat  S )
8 yoneda.o . . . . . . . . . 10  |-  O  =  (oppCat `  C )
9 yoneda.b . . . . . . . . . 10  |-  B  =  ( Base `  C
)
108, 9oppcbas 14653 . . . . . . . . 9  |-  B  =  ( Base `  O
)
11 eqid 2441 . . . . . . . . 9  |-  (comp `  S )  =  (comp `  S )
12 eqid 2441 . . . . . . . . 9  |-  (comp `  Q )  =  (comp `  Q )
13 eqid 2441 . . . . . . . . . . . 12  |-  ( Hom  `  C )  =  ( Hom  `  C )
146, 7fuchom 14867 . . . . . . . . . . . 12  |-  ( O Nat 
S )  =  ( Hom  `  Q )
15 relfunc 14768 . . . . . . . . . . . . 13  |-  Rel  ( C  Func  Q )
16 yoneda.y . . . . . . . . . . . . . 14  |-  Y  =  (Yon `  C )
17 yoneda.c . . . . . . . . . . . . . 14  |-  ( ph  ->  C  e.  Cat )
18 yoneda.s . . . . . . . . . . . . . 14  |-  S  =  ( SetCat `  U )
19 yoneda.w . . . . . . . . . . . . . . 15  |-  ( ph  ->  V  e.  W )
20 yoneda.v . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ran  ( Hom f  `  Q )  u.  U
)  C_  V )
2120unssbd 3531 . . . . . . . . . . . . . . 15  |-  ( ph  ->  U  C_  V )
2219, 21ssexd 4436 . . . . . . . . . . . . . 14  |-  ( ph  ->  U  e.  _V )
23 yoneda.u . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
2416, 17, 8, 18, 6, 22, 23yoncl 15068 . . . . . . . . . . . . 13  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
25 1st2ndbr 6622 . . . . . . . . . . . . 13  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  ( 1st `  Y ) ( C  Func  Q )
( 2nd `  Y
) )
2615, 24, 25sylancr 658 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
27 yonedalem22.p . . . . . . . . . . . 12  |-  ( ph  ->  P  e.  B )
28 yonedalem21.x . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  B )
299, 13, 14, 26, 27, 28funcf2 14774 . . . . . . . . . . 11  |-  ( ph  ->  ( P ( 2nd `  Y ) X ) : ( P ( Hom  `  C ) X ) --> ( ( ( 1st `  Y
) `  P )
( O Nat  S ) ( ( 1st `  Y
) `  X )
) )
30 yonedalem22.k . . . . . . . . . . 11  |-  ( ph  ->  K  e.  ( P ( Hom  `  C
) X ) )
3129, 30ffvelrnd 5841 . . . . . . . . . 10  |-  ( ph  ->  ( ( P ( 2nd `  Y ) X ) `  K
)  e.  ( ( ( 1st `  Y
) `  P )
( O Nat  S ) ( ( 1st `  Y
) `  X )
) )
3231adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( P ( 2nd `  Y
) X ) `  K )  e.  ( ( ( 1st `  Y
) `  P )
( O Nat  S ) ( ( 1st `  Y
) `  X )
) )
33 simpr 458 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F ) )
34 yonedalem22.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  ( F ( O Nat  S ) G ) )
3534adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  A  e.  ( F ( O Nat  S
) G ) )
366, 7, 12, 33, 35fuccocl 14870 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( A (
<. ( ( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) a )  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) G ) )
3727adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  P  e.  B
)
386, 7, 10, 11, 12, 32, 36, 37fuccoval 14869 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( A ( <. (
( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) a ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) `
 P )  =  ( ( ( A ( <. ( ( 1st `  Y ) `  X
) ,  F >. (comp `  Q ) G ) a ) `  P
) ( <. (
( 1st `  (
( 1st `  Y
) `  P )
) `  P ) ,  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  P ) >. (comp `  S )
( ( 1st `  G
) `  P )
) ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) ) )
396, 7, 10, 11, 12, 33, 35, 37fuccoval 14869 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( A ( <. ( ( 1st `  Y ) `  X
) ,  F >. (comp `  Q ) G ) a ) `  P
)  =  ( ( A `  P ) ( <. ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  P ) ,  ( ( 1st `  F ) `  P
) >. (comp `  S
) ( ( 1st `  G ) `  P
) ) ( a `
 P ) ) )
4022adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  U  e.  _V )
41 eqid 2441 . . . . . . . . . . . . . . 15  |-  ( Base `  S )  =  (
Base `  S )
42 relfunc 14768 . . . . . . . . . . . . . . . 16  |-  Rel  ( O  Func  S )
436fucbas 14866 . . . . . . . . . . . . . . . . . 18  |-  ( O 
Func  S )  =  (
Base `  Q )
449, 43, 26funcf1 14772 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 1st `  Y
) : B --> ( O 
Func  S ) )
4544, 28ffvelrnd 5841 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( 1st `  Y
) `  X )  e.  ( O  Func  S
) )
46 1st2ndbr 6622 . . . . . . . . . . . . . . . 16  |-  ( ( Rel  ( O  Func  S )  /\  ( ( 1st `  Y ) `
 X )  e.  ( O  Func  S
) )  ->  ( 1st `  ( ( 1st `  Y ) `  X
) ) ( O 
Func  S ) ( 2nd `  ( ( 1st `  Y
) `  X )
) )
4742, 45, 46sylancr 658 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
) ( O  Func  S ) ( 2nd `  (
( 1st `  Y
) `  X )
) )
4810, 41, 47funcf1 14772 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
) : B --> ( Base `  S ) )
49 eqidd 2442 . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  =  B )
5018, 22setcbas 14942 . . . . . . . . . . . . . . 15  |-  ( ph  ->  U  =  ( Base `  S ) )
5149, 50feq23d 5551 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) : B --> U  <->  ( 1st `  ( ( 1st `  Y
) `  X )
) : B --> ( Base `  S ) ) )
5248, 51mpbird 232 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
) : B --> U )
5352, 27ffvelrnd 5841 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )  e.  U )
5453adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  P )  e.  U )
55 yonedalem21.f . . . . . . . . . . . . . . . 16  |-  ( ph  ->  F  e.  ( O 
Func  S ) )
56 1st2ndbr 6622 . . . . . . . . . . . . . . . 16  |-  ( ( Rel  ( O  Func  S )  /\  F  e.  ( O  Func  S
) )  ->  ( 1st `  F ) ( O  Func  S )
( 2nd `  F
) )
5742, 55, 56sylancr 658 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 1st `  F
) ( O  Func  S ) ( 2nd `  F
) )
5810, 41, 57funcf1 14772 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  S ) )
5949, 50feq23d 5551 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 1st `  F
) : B --> U  <->  ( 1st `  F ) : B --> ( Base `  S )
) )
6058, 59mpbird 232 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1st `  F
) : B --> U )
6160, 27ffvelrnd 5841 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1st `  F
) `  P )  e.  U )
6261adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( 1st `  F ) `  P
)  e.  U )
63 yonedalem22.g . . . . . . . . . . . . . . . 16  |-  ( ph  ->  G  e.  ( O 
Func  S ) )
64 1st2ndbr 6622 . . . . . . . . . . . . . . . 16  |-  ( ( Rel  ( O  Func  S )  /\  G  e.  ( O  Func  S
) )  ->  ( 1st `  G ) ( O  Func  S )
( 2nd `  G
) )
6542, 63, 64sylancr 658 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 1st `  G
) ( O  Func  S ) ( 2nd `  G
) )
6610, 41, 65funcf1 14772 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1st `  G
) : B --> ( Base `  S ) )
6766, 27ffvelrnd 5841 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 1st `  G
) `  P )  e.  ( Base `  S
) )
6867, 50eleqtrrd 2518 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1st `  G
) `  P )  e.  U )
6968adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( 1st `  G ) `  P
)  e.  U )
707, 33nat1st2nd 14857 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  a  e.  (
<. ( 1st `  (
( 1st `  Y
) `  X )
) ,  ( 2nd `  ( ( 1st `  Y
) `  X )
) >. ( O Nat  S
) <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
)
71 eqid 2441 . . . . . . . . . . . . 13  |-  ( Hom  `  S )  =  ( Hom  `  S )
727, 70, 10, 71, 37natcl 14859 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( a `  P )  e.  ( ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
( Hom  `  S ) ( ( 1st `  F
) `  P )
) )
7318, 40, 71, 54, 62elsetchom 14945 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( a `
 P )  e.  ( ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  P )
( Hom  `  S ) ( ( 1st `  F
) `  P )
)  <->  ( a `  P ) : ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P ) --> ( ( 1st `  F
) `  P )
) )
7472, 73mpbid 210 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( a `  P ) : ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P ) --> ( ( 1st `  F
) `  P )
)
757, 34nat1st2nd 14857 . . . . . . . . . . . . . 14  |-  ( ph  ->  A  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. ( O Nat  S ) <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
767, 75, 10, 71, 27natcl 14859 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A `  P
)  e.  ( ( ( 1st `  F
) `  P )
( Hom  `  S ) ( ( 1st `  G
) `  P )
) )
7718, 22, 71, 61, 68elsetchom 14945 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( A `  P )  e.  ( ( ( 1st `  F
) `  P )
( Hom  `  S ) ( ( 1st `  G
) `  P )
)  <->  ( A `  P ) : ( ( 1st `  F
) `  P ) --> ( ( 1st `  G
) `  P )
) )
7876, 77mpbid 210 . . . . . . . . . . . 12  |-  ( ph  ->  ( A `  P
) : ( ( 1st `  F ) `
 P ) --> ( ( 1st `  G
) `  P )
)
7978adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( A `  P ) : ( ( 1st `  F
) `  P ) --> ( ( 1st `  G
) `  P )
)
8018, 40, 11, 54, 62, 69, 74, 79setcco 14947 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( A `
 P ) (
<. ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P ) ,  ( ( 1st `  F ) `  P
) >. (comp `  S
) ( ( 1st `  G ) `  P
) ) ( a `
 P ) )  =  ( ( A `
 P )  o.  ( a `  P
) ) )
8139, 80eqtrd 2473 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( A ( <. ( ( 1st `  Y ) `  X
) ,  F >. (comp `  Q ) G ) a ) `  P
)  =  ( ( A `  P )  o.  ( a `  P ) ) )
8281oveq1d 6105 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( A ( <. (
( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) a ) `  P ) ( <. ( ( 1st `  ( ( 1st `  Y
) `  P )
) `  P ) ,  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  P ) >. (comp `  S )
( ( 1st `  G
) `  P )
) ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) )  =  ( ( ( A `  P )  o.  (
a `  P )
) ( <. (
( 1st `  (
( 1st `  Y
) `  P )
) `  P ) ,  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  P ) >. (comp `  S )
( ( 1st `  G
) `  P )
) ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) ) )
8344, 27ffvelrnd 5841 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 1st `  Y
) `  P )  e.  ( O  Func  S
) )
84 1st2ndbr 6622 . . . . . . . . . . . . . 14  |-  ( ( Rel  ( O  Func  S )  /\  ( ( 1st `  Y ) `
 P )  e.  ( O  Func  S
) )  ->  ( 1st `  ( ( 1st `  Y ) `  P
) ) ( O 
Func  S ) ( 2nd `  ( ( 1st `  Y
) `  P )
) )
8542, 83, 84sylancr 658 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  P )
) ( O  Func  S ) ( 2nd `  (
( 1st `  Y
) `  P )
) )
8610, 41, 85funcf1 14772 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  P )
) : B --> ( Base `  S ) )
8786, 27ffvelrnd 5841 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  P )
) `  P )  e.  ( Base `  S
) )
8887, 50eleqtrrd 2518 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  P )
) `  P )  e.  U )
8988adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( 1st `  ( ( 1st `  Y
) `  P )
) `  P )  e.  U )
907, 31nat1st2nd 14857 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( P ( 2nd `  Y ) X ) `  K
)  e.  ( <.
( 1st `  (
( 1st `  Y
) `  P )
) ,  ( 2nd `  ( ( 1st `  Y
) `  P )
) >. ( O Nat  S
) <. ( 1st `  (
( 1st `  Y
) `  X )
) ,  ( 2nd `  ( ( 1st `  Y
) `  X )
) >. ) )
917, 90, 10, 71, 27natcl 14859 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
)  e.  ( ( ( 1st `  (
( 1st `  Y
) `  P )
) `  P )
( Hom  `  S ) ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
) )
9218, 22, 71, 88, 53elsetchom 14945 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
)  e.  ( ( ( 1st `  (
( 1st `  Y
) `  P )
) `  P )
( Hom  `  S ) ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
)  <->  ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) : ( ( 1st `  ( ( 1st `  Y ) `
 P ) ) `
 P ) --> ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
) )
9391, 92mpbid 210 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) : ( ( 1st `  ( ( 1st `  Y ) `
 P ) ) `
 P ) --> ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
)
9493adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) : ( ( 1st `  ( ( 1st `  Y ) `
 P ) ) `
 P ) --> ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
)
95 fco 5565 . . . . . . . . . 10  |-  ( ( ( A `  P
) : ( ( 1st `  F ) `
 P ) --> ( ( 1st `  G
) `  P )  /\  ( a `  P
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 P ) --> ( ( 1st `  F
) `  P )
)  ->  ( ( A `  P )  o.  ( a `  P
) ) : ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P ) --> ( ( 1st `  G
) `  P )
)
9679, 74, 95syl2anc 656 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( A `
 P )  o.  ( a `  P
) ) : ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P ) --> ( ( 1st `  G
) `  P )
)
9718, 40, 11, 89, 54, 69, 94, 96setcco 14947 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( A `  P )  o.  ( a `  P ) ) (
<. ( ( 1st `  (
( 1st `  Y
) `  P )
) `  P ) ,  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  P ) >. (comp `  S )
( ( 1st `  G
) `  P )
) ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) )  =  ( ( ( A `  P )  o.  (
a `  P )
)  o.  ( ( ( P ( 2nd `  Y ) X ) `
 K ) `  P ) ) )
9838, 82, 973eqtrd 2477 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( A ( <. (
( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) a ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) `
 P )  =  ( ( ( A `
 P )  o.  ( a `  P
) )  o.  (
( ( P ( 2nd `  Y ) X ) `  K
) `  P )
) )
9998fveq1d 5690 . . . . . 6  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( ( A ( <.
( ( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) a ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) `
 P ) `  (  .1.  `  P )
)  =  ( ( ( ( A `  P )  o.  (
a `  P )
)  o.  ( ( ( P ( 2nd `  Y ) X ) `
 K ) `  P ) ) `  (  .1.  `  P )
) )
100 yoneda.1 . . . . . . . . . 10  |-  .1.  =  ( Id `  C )
1019, 13, 100, 17, 27catidcl 14616 . . . . . . . . 9  |-  ( ph  ->  (  .1.  `  P
)  e.  ( P ( Hom  `  C
) P ) )
10216, 9, 17, 27, 13, 27yon11 15070 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  P )
) `  P )  =  ( P ( Hom  `  C ) P ) )
103101, 102eleqtrrd 2518 . . . . . . . 8  |-  ( ph  ->  (  .1.  `  P
)  e.  ( ( 1st `  ( ( 1st `  Y ) `
 P ) ) `
 P ) )
104103adantr 462 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  (  .1.  `  P )  e.  ( ( 1st `  (
( 1st `  Y
) `  P )
) `  P )
)
105 fvco3 5765 . . . . . . 7  |-  ( ( ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) : ( ( 1st `  ( ( 1st `  Y ) `
 P ) ) `
 P ) --> ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )  /\  (  .1.  `  P
)  e.  ( ( 1st `  ( ( 1st `  Y ) `
 P ) ) `
 P ) )  ->  ( ( ( ( A `  P
)  o.  ( a `
 P ) )  o.  ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) ) `  (  .1.  `  P ) )  =  ( ( ( A `  P )  o.  ( a `  P ) ) `  ( ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) `  (  .1.  `  P ) ) ) )
10694, 104, 105syl2anc 656 . . . . . 6  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( ( A `  P
)  o.  ( a `
 P ) )  o.  ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) ) `  (  .1.  `  P ) )  =  ( ( ( A `  P )  o.  ( a `  P ) ) `  ( ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) `  (  .1.  `  P ) ) ) )
10794, 104ffvelrnd 5841 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( ( P ( 2nd `  Y ) X ) `
 K ) `  P ) `  (  .1.  `  P ) )  e.  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  P )
)
108 fvco3 5765 . . . . . . . 8  |-  ( ( ( a `  P
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 P ) --> ( ( 1st `  F
) `  P )  /\  ( ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) `  (  .1.  `  P ) )  e.  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
)  ->  ( (
( A `  P
)  o.  ( a `
 P ) ) `
 ( ( ( ( P ( 2nd `  Y ) X ) `
 K ) `  P ) `  (  .1.  `  P ) ) )  =  ( ( A `  P ) `
 ( ( a `
 P ) `  ( ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) `  (  .1.  `  P ) ) ) ) )
10974, 107, 108syl2anc 656 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( A `  P )  o.  ( a `  P ) ) `  ( ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) `  (  .1.  `  P ) ) )  =  ( ( A `
 P ) `  ( ( a `  P ) `  (
( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) `  (  .1.  `  P ) ) ) ) )
11017adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  C  e.  Cat )
11128adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  X  e.  B
)
112 eqid 2441 . . . . . . . . . . . 12  |-  (comp `  C )  =  (comp `  C )
11330adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  K  e.  ( P ( Hom  `  C
) X ) )
114101adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  (  .1.  `  P )  e.  ( P ( Hom  `  C
) P ) )
11516, 9, 110, 37, 13, 111, 112, 37, 113, 114yon2 15072 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( ( P ( 2nd `  Y ) X ) `
 K ) `  P ) `  (  .1.  `  P ) )  =  ( K (
<. P ,  P >. (comp `  C ) X ) (  .1.  `  P
) ) )
1169, 13, 100, 110, 37, 112, 111, 113catrid 14618 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( K (
<. P ,  P >. (comp `  C ) X ) (  .1.  `  P
) )  =  K )
117115, 116eqtrd 2473 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( ( P ( 2nd `  Y ) X ) `
 K ) `  P ) `  (  .1.  `  P ) )  =  K )
118117fveq2d 5692 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( a `
 P ) `  ( ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) `  (  .1.  `  P ) ) )  =  ( ( a `
 P ) `  K ) )
119 eqid 2441 . . . . . . . . . . . . . . 15  |-  ( Hom  `  O )  =  ( Hom  `  O )
12010, 119, 71, 47, 28, 27funcf2 14774 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( X ( 2nd `  ( ( 1st `  Y
) `  X )
) P ) : ( X ( Hom  `  O ) P ) --> ( ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  X )
( Hom  `  S ) ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
) )
12113, 8oppchom 14650 . . . . . . . . . . . . . . 15  |-  ( X ( Hom  `  O
) P )  =  ( P ( Hom  `  C ) X )
12230, 121syl6eleqr 2532 . . . . . . . . . . . . . 14  |-  ( ph  ->  K  e.  ( X ( Hom  `  O
) P ) )
123120, 122ffvelrnd 5841 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( X ( 2nd `  ( ( 1st `  Y ) `
 X ) ) P ) `  K
)  e.  ( ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X )
( Hom  `  S ) ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
) )
12452, 28ffvelrnd 5841 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X )  e.  U )
12518, 22, 71, 124, 53elsetchom 14945 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( X ( 2nd `  (
( 1st `  Y
) `  X )
) P ) `  K )  e.  ( ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X )
( Hom  `  S ) ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
)  <->  ( ( X ( 2nd `  (
( 1st `  Y
) `  X )
) P ) `  K ) : ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X ) --> ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
) )
126123, 125mpbid 210 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( X ( 2nd `  ( ( 1st `  Y ) `
 X ) ) P ) `  K
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 X ) --> ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
)
127126adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( X ( 2nd `  (
( 1st `  Y
) `  X )
) P ) `  K ) : ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X ) --> ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
)
1289, 13, 100, 17, 28catidcl 14616 . . . . . . . . . . . . 13  |-  ( ph  ->  (  .1.  `  X
)  e.  ( X ( Hom  `  C
) X ) )
12916, 9, 17, 28, 13, 28yon11 15070 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X )  =  ( X ( Hom  `  C ) X ) )
130128, 129eleqtrrd 2518 . . . . . . . . . . . 12  |-  ( ph  ->  (  .1.  `  X
)  e.  ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 X ) )
131130adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  (  .1.  `  X )  e.  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X )
)
132 fvco3 5765 . . . . . . . . . . 11  |-  ( ( ( ( X ( 2nd `  ( ( 1st `  Y ) `
 X ) ) P ) `  K
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 X ) --> ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )  /\  (  .1.  `  X
)  e.  ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 X ) )  ->  ( ( ( a `  P )  o.  ( ( X ( 2nd `  (
( 1st `  Y
) `  X )
) P ) `  K ) ) `  (  .1.  `  X )
)  =  ( ( a `  P ) `
 ( ( ( X ( 2nd `  (
( 1st `  Y
) `  X )
) P ) `  K ) `  (  .1.  `  X ) ) ) )
133127, 131, 132syl2anc 656 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( a `  P )  o.  ( ( X ( 2nd `  (
( 1st `  Y
) `  X )
) P ) `  K ) ) `  (  .1.  `  X )
)  =  ( ( a `  P ) `
 ( ( ( X ( 2nd `  (
( 1st `  Y
) `  X )
) P ) `  K ) `  (  .1.  `  X ) ) ) )
134122adantr 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  K  e.  ( X ( Hom  `  O
) P ) )
1357, 70, 10, 119, 11, 111, 37, 134nati 14861 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( a `
 P ) (
<. ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X ) ,  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  P ) >. (comp `  S )
( ( 1st `  F
) `  P )
) ( ( X ( 2nd `  (
( 1st `  Y
) `  X )
) P ) `  K ) )  =  ( ( ( X ( 2nd `  F
) P ) `  K ) ( <.
( ( 1st `  (
( 1st `  Y
) `  X )
) `  X ) ,  ( ( 1st `  F ) `  X
) >. (comp `  S
) ( ( 1st `  F ) `  P
) ) ( a `
 X ) ) )
136124adantr 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  X )  e.  U )
13718, 40, 11, 136, 54, 62, 127, 74setcco 14947 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( a `
 P ) (
<. ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X ) ,  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  P ) >. (comp `  S )
( ( 1st `  F
) `  P )
) ( ( X ( 2nd `  (
( 1st `  Y
) `  X )
) P ) `  K ) )  =  ( ( a `  P )  o.  (
( X ( 2nd `  ( ( 1st `  Y
) `  X )
) P ) `  K ) ) )
13860, 28ffvelrnd 5841 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 1st `  F
) `  X )  e.  U )
139138adantr 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( 1st `  F ) `  X
)  e.  U )
1407, 70, 10, 71, 111natcl 14859 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( a `  X )  e.  ( ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X )
( Hom  `  S ) ( ( 1st `  F
) `  X )
) )
14118, 40, 71, 136, 139elsetchom 14945 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( a `
 X )  e.  ( ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  X )
( Hom  `  S ) ( ( 1st `  F
) `  X )
)  <->  ( a `  X ) : ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X ) --> ( ( 1st `  F
) `  X )
) )
142140, 141mpbid 210 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( a `  X ) : ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X ) --> ( ( 1st `  F
) `  X )
)
14310, 119, 71, 57, 28, 27funcf2 14774 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( X ( 2nd `  F ) P ) : ( X ( Hom  `  O ) P ) --> ( ( ( 1st `  F
) `  X )
( Hom  `  S ) ( ( 1st `  F
) `  P )
) )
144143, 122ffvelrnd 5841 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( X ( 2nd `  F ) P ) `  K
)  e.  ( ( ( 1st `  F
) `  X )
( Hom  `  S ) ( ( 1st `  F
) `  P )
) )
14518, 22, 71, 138, 61elsetchom 14945 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( ( X ( 2nd `  F
) P ) `  K )  e.  ( ( ( 1st `  F
) `  X )
( Hom  `  S ) ( ( 1st `  F
) `  P )
)  <->  ( ( X ( 2nd `  F
) P ) `  K ) : ( ( 1st `  F
) `  X ) --> ( ( 1st `  F
) `  P )
) )
146144, 145mpbid 210 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( X ( 2nd `  F ) P ) `  K
) : ( ( 1st `  F ) `
 X ) --> ( ( 1st `  F
) `  P )
)
147146adantr 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( X ( 2nd `  F
) P ) `  K ) : ( ( 1st `  F
) `  X ) --> ( ( 1st `  F
) `  P )
)
14818, 40, 11, 136, 139, 62, 142, 147setcco 14947 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( X ( 2nd `  F
) P ) `  K ) ( <.
( ( 1st `  (
( 1st `  Y
) `  X )
) `  X ) ,  ( ( 1st `  F ) `  X
) >. (comp `  S
) ( ( 1st `  F ) `  P
) ) ( a `
 X ) )  =  ( ( ( X ( 2nd `  F
) P ) `  K )  o.  (
a `  X )
) )
149135, 137, 1483eqtr3d 2481 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( a `
 P )  o.  ( ( X ( 2nd `  ( ( 1st `  Y ) `
 X ) ) P ) `  K
) )  =  ( ( ( X ( 2nd `  F ) P ) `  K
)  o.  ( a `
 X ) ) )
150149fveq1d 5690 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( a `  P )  o.  ( ( X ( 2nd `  (
( 1st `  Y
) `  X )
) P ) `  K ) ) `  (  .1.  `  X )
)  =  ( ( ( ( X ( 2nd `  F ) P ) `  K
)  o.  ( a `
 X ) ) `
 (  .1.  `  X ) ) )
151128adantr 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  (  .1.  `  X )  e.  ( X ( Hom  `  C
) X ) )
15216, 9, 110, 111, 13, 111, 112, 37, 113, 151yon12 15071 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( X ( 2nd `  (
( 1st `  Y
) `  X )
) P ) `  K ) `  (  .1.  `  X ) )  =  ( (  .1.  `  X ) ( <. P ,  X >. (comp `  C ) X ) K ) )
1539, 13, 100, 110, 37, 112, 111, 113catlid 14617 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( (  .1.  `  X ) ( <. P ,  X >. (comp `  C ) X ) K )  =  K )
154152, 153eqtrd 2473 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( X ( 2nd `  (
( 1st `  Y
) `  X )
) P ) `  K ) `  (  .1.  `  X ) )  =  K )
155154fveq2d 5692 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( a `
 P ) `  ( ( ( X ( 2nd `  (
( 1st `  Y
) `  X )
) P ) `  K ) `  (  .1.  `  X ) ) )  =  ( ( a `  P ) `
 K ) )
156133, 150, 1553eqtr3d 2481 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( ( X ( 2nd `  F ) P ) `
 K )  o.  ( a `  X
) ) `  (  .1.  `  X ) )  =  ( ( a `
 P ) `  K ) )
157 fvco3 5765 . . . . . . . . . 10  |-  ( ( ( a `  X
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 X ) --> ( ( 1st `  F
) `  X )  /\  (  .1.  `  X
)  e.  ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 X ) )  ->  ( ( ( ( X ( 2nd `  F ) P ) `
 K )  o.  ( a `  X
) ) `  (  .1.  `  X ) )  =  ( ( ( X ( 2nd `  F
) P ) `  K ) `  (
( a `  X
) `  (  .1.  `  X ) ) ) )
158142, 131, 157syl2anc 656 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( ( X ( 2nd `  F ) P ) `
 K )  o.  ( a `  X
) ) `  (  .1.  `  X ) )  =  ( ( ( X ( 2nd `  F
) P ) `  K ) `  (
( a `  X
) `  (  .1.  `  X ) ) ) )
159118, 156, 1583eqtr2d 2479 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( a `
 P ) `  ( ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) `  (  .1.  `  P ) ) )  =  ( ( ( X ( 2nd `  F
) P ) `  K ) `  (
( a `  X
) `  (  .1.  `  X ) ) ) )
160159fveq2d 5692 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( A `
 P ) `  ( ( a `  P ) `  (
( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) `  (  .1.  `  P ) ) ) )  =  ( ( A `  P ) `
 ( ( ( X ( 2nd `  F
) P ) `  K ) `  (
( a `  X
) `  (  .1.  `  X ) ) ) ) )
161109, 160eqtrd 2473 . . . . . 6  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( A `  P )  o.  ( a `  P ) ) `  ( ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) `  (  .1.  `  P ) ) )  =  ( ( A `
 P ) `  ( ( ( X ( 2nd `  F
) P ) `  K ) `  (
( a `  X
) `  (  .1.  `  X ) ) ) ) )
16299, 106, 1613eqtrd 2477 . . . . 5  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( ( A ( <.
( ( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) a ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) `
 P ) `  (  .1.  `  P )
)  =  ( ( A `  P ) `
 ( ( ( X ( 2nd `  F
) P ) `  K ) `  (
( a `  X
) `  (  .1.  `  X ) ) ) ) )
163162mpteq2dva 4375 . . . 4  |-  ( ph  ->  ( a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  |->  ( ( ( ( A (
<. ( ( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) a ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) `
 P ) `  (  .1.  `  P )
) )  =  ( a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  |->  ( ( A `  P ) `
 ( ( ( X ( 2nd `  F
) P ) `  K ) `  (
( a `  X
) `  (  .1.  `  X ) ) ) ) ) )
1645, 163syl5eq 2485 . . 3  |-  ( ph  ->  ( b  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  |->  ( ( ( ( A (
<. ( ( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) b ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) `
 P ) `  (  .1.  `  P )
) )  =  ( a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  |->  ( ( A `  P ) `
 ( ( ( X ( 2nd `  F
) P ) `  K ) `  (
( a `  X
) `  (  .1.  `  X ) ) ) ) ) )
165 eqid 2441 . . . . . . . . . . 11  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
166165, 43, 10xpcbas 14984 . . . . . . . . . 10  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
167 eqid 2441 . . . . . . . . . 10  |-  ( Hom  `  ( Q  X.c  O ) )  =  ( Hom  `  ( Q  X.c  O ) )
168 eqid 2441 . . . . . . . . . 10  |-  ( Hom  `  T )  =  ( Hom  `  T )
169 relfunc 14768 . . . . . . . . . . 11  |-  Rel  (
( Q  X.c  O ) 
Func  T )
170 yoneda.t . . . . . . . . . . . . 13  |-  T  =  ( SetCat `  V )
171 yoneda.h . . . . . . . . . . . . 13  |-  H  =  (HomF
`  Q )
172 yoneda.r . . . . . . . . . . . . 13  |-  R  =  ( ( Q  X.c  O
) FuncCat  T )
173 yoneda.e . . . . . . . . . . . . 13  |-  E  =  ( O evalF  S )
174 yoneda.z . . . . . . . . . . . . 13  |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )
17516, 9, 100, 8, 18, 170, 6, 171, 172, 173, 174, 17, 19, 23, 20yonedalem1 15078 . . . . . . . . . . . 12  |-  ( ph  ->  ( Z  e.  ( ( Q  X.c  O ) 
Func  T )  /\  E  e.  ( ( Q  X.c  O
)  Func  T )
) )
176175simpld 456 . . . . . . . . . . 11  |-  ( ph  ->  Z  e.  ( ( Q  X.c  O )  Func  T
) )
177 1st2ndbr 6622 . . . . . . . . . . 11  |-  ( ( Rel  ( ( Q  X.c  O )  Func  T
)  /\  Z  e.  ( ( Q  X.c  O
)  Func  T )
)  ->  ( 1st `  Z ) ( ( Q  X.c  O )  Func  T
) ( 2nd `  Z
) )
178169, 176, 177sylancr 658 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  Z
) ( ( Q  X.c  O )  Func  T
) ( 2nd `  Z
) )
179 opelxpi 4867 . . . . . . . . . . 11  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  <. F ,  X >.  e.  ( ( O  Func  S )  X.  B ) )
18055, 28, 179syl2anc 656 . . . . . . . . . 10  |-  ( ph  -> 
<. F ,  X >.  e.  ( ( O  Func  S )  X.  B ) )
181 opelxpi 4867 . . . . . . . . . . 11  |-  ( ( G  e.  ( O 
Func  S )  /\  P  e.  B )  ->  <. G ,  P >.  e.  ( ( O  Func  S )  X.  B ) )
18263, 27, 181syl2anc 656 . . . . . . . . . 10  |-  ( ph  -> 
<. G ,  P >.  e.  ( ( O  Func  S )  X.  B ) )
183166, 167, 168, 178, 180, 182funcf2 14774 . . . . . . . . 9  |-  ( ph  ->  ( <. F ,  X >. ( 2nd `  Z
) <. G ,  P >. ) : ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) --> ( ( ( 1st `  Z ) `  <. F ,  X >. )
( Hom  `  T ) ( ( 1st `  Z
) `  <. G ,  P >. ) ) )
184165, 43, 10, 14, 119, 55, 28, 63, 27, 167xpchom2 14992 . . . . . . . . . . 11  |-  ( ph  ->  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. )  =  ( ( F ( O Nat  S ) G )  X.  ( X ( Hom  `  O
) P ) ) )
185121xpeq2i 4857 . . . . . . . . . . 11  |-  ( ( F ( O Nat  S
) G )  X.  ( X ( Hom  `  O ) P ) )  =  ( ( F ( O Nat  S
) G )  X.  ( P ( Hom  `  C ) X ) )
186184, 185syl6eq 2489 . . . . . . . . . 10  |-  ( ph  ->  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. )  =  ( ( F ( O Nat  S ) G )  X.  ( P ( Hom  `  C
) X ) ) )
187 df-ov 6093 . . . . . . . . . . . . 13  |-  ( F ( 1st `  Z
) X )  =  ( ( 1st `  Z
) `  <. F ,  X >. )
188 df-ov 6093 . . . . . . . . . . . . 13  |-  ( G ( 1st `  Z
) P )  =  ( ( 1st `  Z
) `  <. G ,  P >. )
189187, 188oveq12i 6102 . . . . . . . . . . . 12  |-  ( ( F ( 1st `  Z
) X ) ( Hom  `  T )
( G ( 1st `  Z ) P ) )  =  ( ( ( 1st `  Z
) `  <. F ,  X >. ) ( Hom  `  T ) ( ( 1st `  Z ) `
 <. G ,  P >. ) )
190189eqcomi 2445 . . . . . . . . . . 11  |-  ( ( ( 1st `  Z
) `  <. F ,  X >. ) ( Hom  `  T ) ( ( 1st `  Z ) `
 <. G ,  P >. ) )  =  ( ( F ( 1st `  Z ) X ) ( Hom  `  T
) ( G ( 1st `  Z ) P ) )
191190a1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( 1st `  Z ) `  <. F ,  X >. )
( Hom  `  T ) ( ( 1st `  Z
) `  <. G ,  P >. ) )  =  ( ( F ( 1st `  Z ) X ) ( Hom  `  T ) ( G ( 1st `  Z
) P ) ) )
192186, 191feq23d 5551 . . . . . . . . 9  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  Z
) <. G ,  P >. ) : ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) --> ( ( ( 1st `  Z ) `  <. F ,  X >. )
( Hom  `  T ) ( ( 1st `  Z
) `  <. G ,  P >. ) )  <->  ( <. F ,  X >. ( 2nd `  Z ) <. G ,  P >. ) : ( ( F ( O Nat  S ) G )  X.  ( P ( Hom  `  C
) X ) ) --> ( ( F ( 1st `  Z ) X ) ( Hom  `  T ) ( G ( 1st `  Z
) P ) ) ) )
193183, 192mpbid 210 . . . . . . . 8  |-  ( ph  ->  ( <. F ,  X >. ( 2nd `  Z
) <. G ,  P >. ) : ( ( F ( O Nat  S
) G )  X.  ( P ( Hom  `  C ) X ) ) --> ( ( F ( 1st `  Z
) X ) ( Hom  `  T )
( G ( 1st `  Z ) P ) ) )
194193, 34, 30fovrnd 6234 . . . . . . 7  |-  ( ph  ->  ( A ( <. F ,  X >. ( 2nd `  Z )
<. G ,  P >. ) K )  e.  ( ( F ( 1st `  Z ) X ) ( Hom  `  T
) ( G ( 1st `  Z ) P ) ) )
195 eqid 2441 . . . . . . . . . . 11  |-  ( Base `  T )  =  (
Base `  T )
196166, 195, 178funcf1 14772 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  Z
) : ( ( O  Func  S )  X.  B ) --> ( Base `  T ) )
197196, 55, 28fovrnd 6234 . . . . . . . . 9  |-  ( ph  ->  ( F ( 1st `  Z ) X )  e.  ( Base `  T
) )
198170, 19setcbas 14942 . . . . . . . . 9  |-  ( ph  ->  V  =  ( Base `  T ) )
199197, 198eleqtrrd 2518 . . . . . . . 8  |-  ( ph  ->  ( F ( 1st `  Z ) X )  e.  V )
200196, 63, 27fovrnd 6234 . . . . . . . . 9  |-  ( ph  ->  ( G ( 1st `  Z ) P )  e.  ( Base `  T
) )
201200, 198eleqtrrd 2518 . . . . . . . 8  |-  ( ph  ->  ( G ( 1st `  Z ) P )  e.  V )
202170, 19, 168, 199, 201elsetchom 14945 . . . . . . 7  |-  ( ph  ->  ( ( A (
<. F ,  X >. ( 2nd `  Z )
<. G ,  P >. ) K )  e.  ( ( F ( 1st `  Z ) X ) ( Hom  `  T
) ( G ( 1st `  Z ) P ) )  <->  ( A
( <. F ,  X >. ( 2nd `  Z
) <. G ,  P >. ) K ) : ( F ( 1st `  Z ) X ) --> ( G ( 1st `  Z ) P ) ) )
203194, 202mpbid 210 . . . . . 6  |-  ( ph  ->  ( A ( <. F ,  X >. ( 2nd `  Z )
<. G ,  P >. ) K ) : ( F ( 1st `  Z
) X ) --> ( G ( 1st `  Z
) P ) )
20416, 9, 100, 8, 18, 170, 6, 171, 172, 173, 174, 17, 19, 23, 20, 55, 28, 63, 27, 34, 30yonedalem22 15084 . . . . . . . 8  |-  ( ph  ->  ( A ( <. F ,  X >. ( 2nd `  Z )
<. G ,  P >. ) K )  =  ( ( ( P ( 2nd `  Y ) X ) `  K
) ( <. (
( 1st `  Y
) `  X ) ,  F >. ( 2nd `  H
) <. ( ( 1st `  Y ) `  P
) ,  G >. ) A ) )
2058oppccat 14657 . . . . . . . . . . 11  |-  ( C  e.  Cat  ->  O  e.  Cat )
20617, 205syl 16 . . . . . . . . . 10  |-  ( ph  ->  O  e.  Cat )
20718setccat 14949 . . . . . . . . . . 11  |-  ( U  e.  _V  ->  S  e.  Cat )
20822, 207syl 16 . . . . . . . . . 10  |-  ( ph  ->  S  e.  Cat )
2096, 206, 208fuccat 14876 . . . . . . . . 9  |-  ( ph  ->  Q  e.  Cat )
210171, 209, 43, 14, 45, 55, 83, 63, 12, 31, 34hof2val 15062 . . . . . . . 8  |-  ( ph  ->  ( ( ( P ( 2nd `  Y
) X ) `  K ) ( <.
( ( 1st `  Y
) `  X ) ,  F >. ( 2nd `  H
) <. ( ( 1st `  Y ) `  P
) ,  G >. ) A )  =  ( b  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  |->  ( ( A ( <. (
( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) b ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) ) )
211204, 210eqtrd 2473 . . . . . . 7  |-  ( ph  ->  ( A ( <. F ,  X >. ( 2nd `  Z )
<. G ,  P >. ) K )  =  ( b  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  |->  ( ( A ( <. (
( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) b ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) ) )
21216, 9, 100, 8, 18, 170, 6, 171, 172, 173, 174, 17, 19, 23, 20, 55, 28yonedalem21 15079 . . . . . . 7  |-  ( ph  ->  ( F ( 1st `  Z ) X )  =  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F ) )
21316, 9, 100, 8, 18, 170, 6, 171, 172, 173, 174, 17, 19, 23, 20, 63, 27yonedalem21 15079 . . . . . . 7  |-  ( ph  ->  ( G ( 1st `  Z ) P )  =  ( ( ( 1st `  Y ) `
 P ) ( O Nat  S ) G ) )
214211, 212, 213feq123d 5546 . . . . . 6  |-  ( ph  ->  ( ( A (
<. F ,  X >. ( 2nd `  Z )
<. G ,  P >. ) K ) : ( F ( 1st `  Z
) X ) --> ( G ( 1st `  Z
) P )  <->  ( b  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F )  |->  ( ( A ( <.
( ( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) b ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) ) : ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F ) --> ( ( ( 1st `  Y
) `  P )
( O Nat  S ) G ) ) )
215203, 214mpbid 210 . . . . 5  |-  ( ph  ->  ( b  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  |->  ( ( A ( <. (
( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) b ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) ) : ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F ) --> ( ( ( 1st `  Y
) `  P )
( O Nat  S ) G ) )
216 eqid 2441 . . . . . 6  |-  ( b  e.  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F )  |->  ( ( A ( <. ( ( 1st `  Y ) `  X
) ,  F >. (comp `  Q ) G ) b ) ( <.
( ( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) )  =  ( b  e.  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F )  |->  ( ( A ( <. ( ( 1st `  Y ) `  X
) ,  F >. (comp `  Q ) G ) b ) ( <.
( ( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) )
217216fmpt 5861 . . . . 5  |-  ( A. b  e.  ( (
( 1st `  Y
) `  X )
( O Nat  S ) F ) ( ( A ( <. (
( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) b ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) )  e.  ( ( ( 1st `  Y ) `
 P ) ( O Nat  S ) G )  <->  ( b  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F )  |->  ( ( A ( <.
( ( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) b ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) ) : ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F ) --> ( ( ( 1st `  Y
) `  P )
( O Nat  S ) G ) )
218215, 217sylibr 212 . . . 4  |-  ( ph  ->  A. b  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F ) ( ( A ( <. (
( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) b ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) )  e.  ( ( ( 1st `  Y ) `
 P ) ( O Nat  S ) G ) )
219 yonedalem3.m . . . . . 6  |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y
) `  x )
( O Nat  S ) f )  |->  ( ( a `  x ) `
 (  .1.  `  x ) ) ) )
22016, 9, 100, 8, 18, 170, 6, 171, 172, 173, 174, 17, 19, 23, 20, 63, 27, 219yonedalem3a 15080 . . . . 5  |-  ( ph  ->  ( ( G M P )  =  ( a  e.  ( ( ( 1st `  Y
) `  P )
( O Nat  S ) G )  |->  ( ( a `  P ) `
 (  .1.  `  P ) ) )  /\  ( G M P ) : ( G ( 1st `  Z
) P ) --> ( G ( 1st `  E
) P ) ) )
221220simpld 456 . . . 4  |-  ( ph  ->  ( G M P )  =  ( a  e.  ( ( ( 1st `  Y ) `
 P ) ( O Nat  S ) G )  |->  ( ( a `
 P ) `  (  .1.  `  P )
) ) )
222 fveq1 5687 . . . . 5  |-  ( a  =  ( ( A ( <. ( ( 1st `  Y ) `  X
) ,  F >. (comp `  Q ) G ) b ) ( <.
( ( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) )  ->  ( a `  P )  =  ( ( ( A (
<. ( ( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) b ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) `
 P ) )
223222fveq1d 5690 . . . 4  |-  ( a  =  ( ( A ( <. ( ( 1st `  Y ) `  X
) ,  F >. (comp `  Q ) G ) b ) ( <.
( ( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) )  ->  ( ( a `
 P ) `  (  .1.  `  P )
)  =  ( ( ( ( A (
<. ( ( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) b ) ( <. (
( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) `
 P ) `  (  .1.  `  P )
) )
224218, 211, 221, 223fmptcof 5874 . . 3  |-  ( ph  ->  ( ( G M P )  o.  ( A ( <. F ,  X >. ( 2nd `  Z
) <. G ,  P >. ) K ) )  =  ( b  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F )  |->  ( ( ( ( A ( <. ( ( 1st `  Y ) `  X
) ,  F >. (comp `  Q ) G ) b ) ( <.
( ( 1st `  Y
) `  P ) ,  ( ( 1st `  Y ) `  X
) >. (comp `  Q
) G ) ( ( P ( 2nd `  Y ) X ) `
 K ) ) `
 P ) `  (  .1.  `  P )
) ) )
225 eqid 2441 . . . . . . 7  |-  ( <. F ,  X >. ( 2nd `  E )
<. G ,  P >. )  =  ( <. F ,  X >. ( 2nd `  E
) <. G ,  P >. )
226173, 206, 208, 10, 119, 11, 7, 55, 63, 28, 27, 225, 34, 122evlf2val 15025 . . . . . 6  |-  ( ph  ->  ( A ( <. F ,  X >. ( 2nd `  E )
<. G ,  P >. ) K )  =  ( ( A `  P
) ( <. (
( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  P
) >. (comp `  S
) ( ( 1st `  G ) `  P
) ) ( ( X ( 2nd `  F
) P ) `  K ) ) )
22718, 22, 11, 138, 61, 68, 146, 78setcco 14947 . . . . . 6  |-  ( ph  ->  ( ( A `  P ) ( <.
( ( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  P
) >. (comp `  S
) ( ( 1st `  G ) `  P
) ) ( ( X ( 2nd `  F
) P ) `  K ) )  =  ( ( A `  P )  o.  (
( X ( 2nd `  F ) P ) `
 K ) ) )
228226, 227eqtrd 2473 . . . . 5  |-  ( ph  ->  ( A ( <. F ,  X >. ( 2nd `  E )
<. G ,  P >. ) K )  =  ( ( A `  P
)  o.  ( ( X ( 2nd `  F
) P ) `  K ) ) )
229228coeq1d 4997 . . . 4  |-  ( ph  ->  ( ( A (
<. F ,  X >. ( 2nd `  E )
<. G ,  P >. ) K )  o.  ( F M X ) )  =  ( ( ( A `  P )  o.  ( ( X ( 2nd `  F
) P ) `  K ) )  o.  ( F M X ) ) )
23016, 9, 100, 8, 18, 170, 6, 171, 172, 173, 174, 17, 19, 23, 20, 55, 28, 219yonedalem3a 15080 . . . . . . . 8  |-  ( ph  ->  ( ( F M X )  =  ( a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  |->  ( ( a `  X ) `
 (  .1.  `  X ) ) )  /\  ( F M X ) : ( F ( 1st `  Z
) X ) --> ( F ( 1st `  E
) X ) ) )
231230simprd 460 . . . . . . 7  |-  ( ph  ->  ( F M X ) : ( F ( 1st `  Z
) X ) --> ( F ( 1st `  E
) X ) )
232230simpld 456 . . . . . . . 8  |-  ( ph  ->  ( F M X )  =  ( a  e.  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F )  |->  ( ( a `
 X ) `  (  .1.  `  X )
) ) )
233173, 206, 208, 10, 55, 28evlf1 15026 . . . . . . . 8  |-  ( ph  ->  ( F ( 1st `  E ) X )  =  ( ( 1st `  F ) `  X
) )
234232, 212, 233feq123d 5546 . . . . . . 7  |-  ( ph  ->  ( ( F M X ) : ( F ( 1st `  Z
) X ) --> ( F ( 1st `  E
) X )  <->  ( a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F )  |->  ( ( a `  X
) `  (  .1.  `  X ) ) ) : ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F ) --> ( ( 1st `  F ) `  X
) ) )
235231, 234mpbid 210 . . . . . 6  |-  ( ph  ->  ( a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  |->  ( ( a `  X ) `
 (  .1.  `  X ) ) ) : ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F ) --> ( ( 1st `  F ) `  X
) )
236 eqid 2441 . . . . . . 7  |-  ( a  e.  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F )  |->  ( ( a `
 X ) `  (  .1.  `  X )
) )  =  ( a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  |->  ( ( a `  X ) `
 (  .1.  `  X ) ) )
237236fmpt 5861 . . . . . 6  |-  ( A. a  e.  ( (
( 1st `  Y
) `  X )
( O Nat  S ) F ) ( ( a `  X ) `
 (  .1.  `  X ) )  e.  ( ( 1st `  F
) `  X )  <->  ( a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  |->  ( ( a `  X ) `
 (  .1.  `  X ) ) ) : ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F ) --> ( ( 1st `  F ) `  X
) )
238235, 237sylibr 212 . . . . 5  |-  ( ph  ->  A. a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F ) ( ( a `  X ) `
 (  .1.  `  X ) )  e.  ( ( 1st `  F
) `  X )
)
239 fcompt 5876 . . . . . 6  |-  ( ( ( A `  P
) : ( ( 1st `  F ) `
 P ) --> ( ( 1st `  G
) `  P )  /\  ( ( X ( 2nd `  F ) P ) `  K
) : ( ( 1st `  F ) `
 X ) --> ( ( 1st `  F
) `  P )
)  ->  ( ( A `  P )  o.  ( ( X ( 2nd `  F ) P ) `  K
) )  =  ( y  e.  ( ( 1st `  F ) `
 X )  |->  ( ( A `  P
) `  ( (
( X ( 2nd `  F ) P ) `
 K ) `  y ) ) ) )
24078, 146, 239syl2anc 656 . . . . 5  |-  ( ph  ->  ( ( A `  P )  o.  (
( X ( 2nd `  F ) P ) `
 K ) )  =  ( y  e.  ( ( 1st `  F
) `  X )  |->  ( ( A `  P ) `  (
( ( X ( 2nd `  F ) P ) `  K
) `  y )
) ) )
241 fveq2 5688 . . . . . 6  |-  ( y  =  ( ( a `
 X ) `  (  .1.  `  X )
)  ->  ( (
( X ( 2nd `  F ) P ) `
 K ) `  y )  =  ( ( ( X ( 2nd `  F ) P ) `  K
) `  ( (
a `  X ) `  (  .1.  `  X
) ) ) )
242241fveq2d 5692 . . . . 5  |-  ( y  =  ( ( a `
 X ) `  (  .1.  `  X )
)  ->  ( ( A `  P ) `  ( ( ( X ( 2nd `  F
) P ) `  K ) `  y
) )  =  ( ( A `  P
) `  ( (
( X ( 2nd `  F ) P ) `
 K ) `  ( ( a `  X ) `  (  .1.  `  X ) ) ) ) )
243238, 232, 240, 242fmptcof 5874 . . . 4  |-  ( ph  ->  ( ( ( A `
 P )  o.  ( ( X ( 2nd `  F ) P ) `  K
) )  o.  ( F M X ) )  =  ( a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F )  |->  ( ( A `  P
) `  ( (
( X ( 2nd `  F ) P ) `
 K ) `  ( ( a `  X ) `  (  .1.  `  X ) ) ) ) ) )
244229, 243eqtrd 2473 . . 3  |-  ( ph  ->  ( ( A (
<. F ,  X >. ( 2nd `  E )
<. G ,  P >. ) K )  o.  ( F M X ) )  =  ( a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F )  |->  ( ( A `  P
) `  ( (
( X ( 2nd `  F ) P ) `
 K ) `  ( ( a `  X ) `  (  .1.  `  X ) ) ) ) ) )
245164, 224, 2443eqtr4d 2483 . 2  |-  ( ph  ->  ( ( G M P )  o.  ( A ( <. F ,  X >. ( 2nd `  Z
) <. G ,  P >. ) K ) )  =  ( ( A ( <. F ,  X >. ( 2nd `  E
) <. G ,  P >. ) K )  o.  ( F M X ) ) )
246 eqid 2441 . . 3  |-  (comp `  T )  =  (comp `  T )
247175simprd 460 . . . . . . 7  |-  ( ph  ->  E  e.  ( ( Q  X.c  O )  Func  T
) )
248 1st2ndbr 6622 . . . . . . 7  |-  ( ( Rel  ( ( Q  X.c  O )  Func  T
)  /\  E  e.  ( ( Q  X.c  O
)  Func  T )
)  ->  ( 1st `  E ) ( ( Q  X.c  O )  Func  T
) ( 2nd `  E
) )
249169, 247, 248sylancr 658 . . . . . 6  |-  ( ph  ->  ( 1st `  E
) ( ( Q  X.c  O )  Func  T
) ( 2nd `  E
) )
250166, 195, 249funcf1 14772 . . . . 5  |-  ( ph  ->  ( 1st `  E
) : ( ( O  Func  S )  X.  B ) --> ( Base `  T ) )
251250, 63, 27fovrnd 6234 . . . 4  |-  ( ph  ->  ( G ( 1st `  E ) P )  e.  ( Base `  T
) )
252251, 198eleqtrrd 2518 . . 3  |-  ( ph  ->  ( G ( 1st `  E ) P )  e.  V )
253220simprd 460 . . 3  |-  ( ph  ->  ( G M P ) : ( G ( 1st `  Z
) P ) --> ( G ( 1st `  E
) P ) )
254170, 19, 246, 199, 201, 252, 203, 253setcco 14947 . 2  |-  ( ph  ->  ( ( G M P ) ( <.
( F ( 1st `  Z ) X ) ,  ( G ( 1st `  Z ) P ) >. (comp `  T ) ( G ( 1st `  E
) P ) ) ( A ( <. F ,  X >. ( 2nd `  Z )
<. G ,  P >. ) K ) )  =  ( ( G M P )  o.  ( A ( <. F ,  X >. ( 2nd `  Z
) <. G ,  P >. ) K ) ) )
255250, 55, 28fovrnd 6234 . . . 4  |-  ( ph  ->  ( F ( 1st `  E ) X )  e.  ( Base `  T
) )
256255, 198eleqtrrd 2518 . . 3  |-  ( ph  ->  ( F ( 1st `  E ) X )  e.  V )
257166, 167, 168, 249, 180, 182funcf2 14774 . . . . . 6  |-  ( ph  ->  ( <. F ,  X >. ( 2nd `  E
) <. G ,  P >. ) : ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) --> ( ( ( 1st `  E ) `  <. F ,  X >. )
( Hom  `  T ) ( ( 1st `  E
) `  <. G ,  P >. ) ) )
258 df-ov 6093 . . . . . . . . . 10  |-  ( F ( 1st `  E
) X )  =  ( ( 1st `  E
) `  <. F ,  X >. )
259 df-ov 6093 . . . . . . . . . 10  |-  ( G ( 1st `  E
) P )  =  ( ( 1st `  E
) `  <. G ,  P >. )
260258, 259oveq12i 6102 . . . . . . . . 9  |-  ( ( F ( 1st `  E
) X ) ( Hom  `  T )
( G ( 1st `  E ) P ) )  =  ( ( ( 1st `  E
) `  <. F ,  X >. ) ( Hom  `  T ) ( ( 1st `  E ) `
 <. G ,  P >. ) )
261260eqcomi 2445 . . . . . . . 8  |-  ( ( ( 1st `  E
) `  <. F ,  X >. ) ( Hom  `  T ) ( ( 1st `  E ) `
 <. G ,  P >. ) )  =  ( ( F ( 1st `  E ) X ) ( Hom  `  T
) ( G ( 1st `  E ) P ) )
262261a1i 11 . . . . . . 7  |-  ( ph  ->  ( ( ( 1st `  E ) `  <. F ,  X >. )
( Hom  `  T ) ( ( 1st `  E
) `  <. G ,  P >. ) )  =  ( ( F ( 1st `  E ) X ) ( Hom  `  T ) ( G ( 1st `  E
) P ) ) )
263186, 262feq23d 5551 . . . . . 6  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  E
) <. G ,  P >. ) : ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) --> ( ( ( 1st `  E ) `  <. F ,  X >. )
( Hom  `  T ) ( ( 1st `  E
) `  <. G ,  P >. ) )  <->  ( <. F ,  X >. ( 2nd `  E ) <. G ,  P >. ) : ( ( F ( O Nat  S ) G )  X.  ( P ( Hom  `  C
) X ) ) --> ( ( F ( 1st `  E ) X ) ( Hom  `  T ) ( G ( 1st `  E
) P ) ) ) )
264257, 263mpbid 210 . . . . 5  |-  ( ph  ->  ( <. F ,  X >. ( 2nd `  E
) <. G ,  P >. ) : ( ( F ( O Nat  S
) G )  X.  ( P ( Hom  `  C ) X ) ) --> ( ( F ( 1st `  E
) X ) ( Hom  `  T )
( G ( 1st `  E ) P ) ) )
265264, 34, 30fovrnd 6234 . . . 4  |-  ( ph  ->  ( A ( <. F ,  X >. ( 2nd `  E )
<. G ,  P >. ) K )  e.  ( ( F ( 1st `  E ) X ) ( Hom  `  T
) ( G ( 1st `  E ) P ) ) )
266170, 19, 168, 256, 252elsetchom 14945 . . . 4  |-  ( ph  ->  ( ( A (
<. F ,  X >. ( 2nd `  E )
<. G ,  P >. ) K )  e.  ( ( F ( 1st `  E ) X ) ( Hom  `  T
) ( G ( 1st `  E ) P ) )  <->  ( A
( <. F ,  X >. ( 2nd `  E
) <. G ,  P >. ) K ) : ( F ( 1st `  E ) X ) --> ( G ( 1st `  E ) P ) ) )
267265, 266mpbid 210 . . 3  |-  ( ph  ->  ( A ( <. F ,  X >. ( 2nd `  E )
<. G ,  P >. ) K ) : ( F ( 1st `  E
) X ) --> ( G ( 1st `  E
) P ) )
268170, 19, 246, 199, 256, 252, 231, 267setcco 14947 . 2  |-  ( ph  ->  ( ( A (
<. F ,  X >. ( 2nd `  E )
<. G ,  P >. ) K ) ( <.
( F ( 1st `  Z ) X ) ,  ( F ( 1st `  E ) X ) >. (comp `  T ) ( G ( 1st `  E
) P ) ) ( F M X ) )  =  ( ( A ( <. F ,  X >. ( 2nd `  E )
<. G ,  P >. ) K )  o.  ( F M X ) ) )
269245, 254, 2683eqtr4d 2483 1  |-  ( ph  ->  ( ( G M P ) ( <.
( F ( 1st `  Z ) X ) ,  ( G ( 1st `  Z ) P ) >. (comp `  T ) ( G ( 1st `  E
) P ) ) ( A ( <. F ,  X >. ( 2nd `  Z )
<. G ,  P >. ) K ) )  =  ( ( A (
<. F ,  X >. ( 2nd `  E )
<. G ,  P >. ) K ) ( <.
( F ( 1st `  Z ) X ) ,  ( F ( 1st `  E ) X ) >. (comp `  T ) ( G ( 1st `  E
) P ) ) ( F M X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970    u. cun 3323    C_ wss 3325   <.cop 3880   class class class wbr 4289    e. cmpt 4347    X. cxp 4834   ran crn 4837    o. ccom 4840   Rel wrel 4841   -->wf 5411   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   1stc1st 6574   2ndc2nd 6575  tpos ctpos 6743   Basecbs 14170   Hom chom 14245  compcco 14246   Catccat 14598   Idccid 14599   Hom f chomf 14600  oppCatcoppc 14646    Func cfunc 14760    o.func ccofu 14762   Nat cnat 14847   FuncCat cfuc 14848   SetCatcsetc 14939    X.c cxpc 14974    1stF c1stf 14975    2ndF c2ndf 14976   ⟨,⟩F cprf 14977   evalF cevlf 15015  HomFchof 15054  Yoncyon 15055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-hom 14258  df-cco 14259  df-cat 14602  df-cid 14603  df-homf 14604  df-comf 14605  df-oppc 14647  df-ssc 14719  df-resc 14720  df-subc 14721  df-func 14764  df-cofu 14766  df-nat 14849  df-fuc 14850  df-setc 14940  df-xpc 14978  df-1stf 14979  df-2ndf 14980  df-prf 14981  df-evlf 15019  df-curf 15020  df-hof 15056  df-yon 15057
This theorem is referenced by:  yonedalem3  15086
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