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Theorem yonedalem3b 14069
Description: Lemma for yoneda 14073. (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  (  Homf  `  C ) 
C_  U )
yoneda.v  |-  ( ph  ->  ( ran  (  Homf  `  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 5882 . . . . . . . 8  |-  ( b  =  a  ->  ( A ( <. (
( 1st `  Y
) `  X ) ,  F >. (comp `  Q
) G ) b )  =  ( A ( <. ( ( 1st `  Y ) `  X
) ,  F >. (comp `  Q ) G ) a ) )
21oveq1d 5889 . . . . . . 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 5543 . . . . . 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 5543 . . . . 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 4127 . . . 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 2296 . . . . . . . . 9  |-  ( O Nat 
S )  =  ( O Nat  S )
8 yoneda.o . . . . . . . . . 10  |-  O  =  (oppCat `  C )
9 yoneda.b . . . . . . . . . 10  |-  B  =  ( Base `  C
)
108, 9oppcbas 13637 . . . . . . . . 9  |-  B  =  ( Base `  O
)
11 eqid 2296 . . . . . . . . 9  |-  (comp `  S )  =  (comp `  S )
12 eqid 2296 . . . . . . . . 9  |-  (comp `  Q )  =  (comp `  Q )
13 eqid 2296 . . . . . . . . . . . 12  |-  (  Hom  `  C )  =  (  Hom  `  C )
146, 7fuchom 13851 . . . . . . . . . . . 12  |-  ( O Nat 
S )  =  (  Hom  `  Q )
15 relfunc 13752 . . . . . . . . . . . . 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 ssun2 3352 . . . . . . . . . . . . . . . 16  |-  U  C_  ( ran  (  Homf  `  Q )  u.  U )
20 yoneda.v . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U
)  C_  V )
2119, 20syl5ss 3203 . . . . . . . . . . . . . . 15  |-  ( ph  ->  U  C_  V )
22 yoneda.w . . . . . . . . . . . . . . 15  |-  ( ph  ->  V  e.  W )
23 ssexg 4176 . . . . . . . . . . . . . . 15  |-  ( ( U  C_  V  /\  V  e.  W )  ->  U  e.  _V )
2421, 22, 23syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ph  ->  U  e.  _V )
25 yoneda.u . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  (  Homf  `  C ) 
C_  U )
2616, 17, 8, 18, 6, 24, 25yoncl 14052 . . . . . . . . . . . . 13  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
27 1st2ndbr 6185 . . . . . . . . . . . . 13  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  ( 1st `  Y ) ( C  Func  Q )
( 2nd `  Y
) )
2815, 26, 27sylancr 644 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
29 yonedalem22.p . . . . . . . . . . . 12  |-  ( ph  ->  P  e.  B )
30 yonedalem21.x . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  B )
319, 13, 14, 28, 29, 30funcf2 13758 . . . . . . . . . . 11  |-  ( ph  ->  ( P ( 2nd `  Y ) X ) : ( P (  Hom  `  C ) X ) --> ( ( ( 1st `  Y
) `  P )
( O Nat  S ) ( ( 1st `  Y
) `  X )
) )
32 yonedalem22.k . . . . . . . . . . 11  |-  ( ph  ->  K  e.  ( P (  Hom  `  C
) X ) )
3331, 32ffvelrnd 5682 . . . . . . . . . 10  |-  ( ph  ->  ( ( P ( 2nd `  Y ) X ) `  K
)  e.  ( ( ( 1st `  Y
) `  P )
( O Nat  S ) ( ( 1st `  Y
) `  X )
) )
3433adantr 451 . . . . . . . . 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 )
) )
356fucbas 13850 . . . . . . . . . 10  |-  ( O 
Func  S )  =  (
Base `  Q )
368oppccat 13641 . . . . . . . . . . . . 13  |-  ( C  e.  Cat  ->  O  e.  Cat )
3717, 36syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  O  e.  Cat )
3818setccat 13933 . . . . . . . . . . . . 13  |-  ( U  e.  _V  ->  S  e.  Cat )
3924, 38syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  S  e.  Cat )
406, 37, 39fuccat 13860 . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  Cat )
4140adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  Q  e.  Cat )
429, 35, 28funcf1 13756 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st `  Y
) : B --> ( O 
Func  S ) )
4342, 30ffvelrnd 5682 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 1st `  Y
) `  X )  e.  ( O  Func  S
) )
4443adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( 1st `  Y ) `  X
)  e.  ( O 
Func  S ) )
45 yonedalem21.f . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( O 
Func  S ) )
4645adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  F  e.  ( O  Func  S )
)
47 yonedalem22.g . . . . . . . . . . 11  |-  ( ph  ->  G  e.  ( O 
Func  S ) )
4847adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  G  e.  ( O  Func  S )
)
49 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F ) )
50 yonedalem22.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  ( F ( O Nat  S ) G ) )
5150adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  A  e.  ( F ( O Nat  S
) G ) )
5235, 14, 12, 41, 44, 46, 48, 49, 51catcocl 13603 . . . . . . . . 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 ) )
5329adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  P  e.  B
)
546, 7, 10, 11, 12, 34, 52, 53fuccoval 13853 . . . . . . . 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
) ) )
556, 7, 10, 11, 12, 49, 51, 53fuccoval 13853 . . . . . . . . . 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 ) ) )
5624adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  U  e.  _V )
57 eqid 2296 . . . . . . . . . . . . . . 15  |-  ( Base `  S )  =  (
Base `  S )
58 relfunc 13752 . . . . . . . . . . . . . . . 16  |-  Rel  ( O  Func  S )
59 1st2ndbr 6185 . . . . . . . . . . . . . . . 16  |-  ( ( Rel  ( O  Func  S )  /\  ( ( 1st `  Y ) `
 X )  e.  ( O  Func  S
) )  ->  ( 1st `  ( ( 1st `  Y ) `  X
) ) ( O 
Func  S ) ( 2nd `  ( ( 1st `  Y
) `  X )
) )
6058, 43, 59sylancr 644 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
) ( O  Func  S ) ( 2nd `  (
( 1st `  Y
) `  X )
) )
6110, 57, 60funcf1 13756 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
) : B --> ( Base `  S ) )
62 eqidd 2297 . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  =  B )
6318, 24setcbas 13926 . . . . . . . . . . . . . . 15  |-  ( ph  ->  U  =  ( Base `  S ) )
6462, 63feq23d 5402 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) : B --> U  <->  ( 1st `  ( ( 1st `  Y
) `  X )
) : B --> ( Base `  S ) ) )
6561, 64mpbird 223 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
) : B --> U )
6665, 29ffvelrnd 5682 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )  e.  U )
6766adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  P )  e.  U )
68 1st2ndbr 6185 . . . . . . . . . . . . . . . 16  |-  ( ( Rel  ( O  Func  S )  /\  F  e.  ( O  Func  S
) )  ->  ( 1st `  F ) ( O  Func  S )
( 2nd `  F
) )
6958, 45, 68sylancr 644 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 1st `  F
) ( O  Func  S ) ( 2nd `  F
) )
7010, 57, 69funcf1 13756 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  S ) )
7162, 63feq23d 5402 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 1st `  F
) : B --> U  <->  ( 1st `  F ) : B --> ( Base `  S )
) )
7270, 71mpbird 223 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1st `  F
) : B --> U )
7372, 29ffvelrnd 5682 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1st `  F
) `  P )  e.  U )
7473adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( 1st `  F ) `  P
)  e.  U )
75 1st2ndbr 6185 . . . . . . . . . . . . . . . 16  |-  ( ( Rel  ( O  Func  S )  /\  G  e.  ( O  Func  S
) )  ->  ( 1st `  G ) ( O  Func  S )
( 2nd `  G
) )
7658, 47, 75sylancr 644 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 1st `  G
) ( O  Func  S ) ( 2nd `  G
) )
7710, 57, 76funcf1 13756 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1st `  G
) : B --> ( Base `  S ) )
7877, 29ffvelrnd 5682 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 1st `  G
) `  P )  e.  ( Base `  S
) )
7978, 63eleqtrrd 2373 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1st `  G
) `  P )  e.  U )
8079adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( 1st `  G ) `  P
)  e.  U )
817, 49nat1st2nd 13841 . . . . . . . . . . . . 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 ) >. )
)
82 eqid 2296 . . . . . . . . . . . . 13  |-  (  Hom  `  S )  =  (  Hom  `  S )
837, 81, 10, 82, 53natcl 13843 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( a `  P )  e.  ( ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
(  Hom  `  S ) ( ( 1st `  F
) `  P )
) )
8418, 56, 82, 67, 74elsetchom 13929 . . . . . . . . . . . 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 )
) )
8583, 84mpbid 201 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( a `  P ) : ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P ) --> ( ( 1st `  F
) `  P )
)
867, 50nat1st2nd 13841 . . . . . . . . . . . . . 14  |-  ( ph  ->  A  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. ( O Nat  S ) <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
877, 86, 10, 82, 29natcl 13843 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A `  P
)  e.  ( ( ( 1st `  F
) `  P )
(  Hom  `  S ) ( ( 1st `  G
) `  P )
) )
8818, 24, 82, 73, 79elsetchom 13929 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( A `  P )  e.  ( ( ( 1st `  F
) `  P )
(  Hom  `  S ) ( ( 1st `  G
) `  P )
)  <->  ( A `  P ) : ( ( 1st `  F
) `  P ) --> ( ( 1st `  G
) `  P )
) )
8987, 88mpbid 201 . . . . . . . . . . . 12  |-  ( ph  ->  ( A `  P
) : ( ( 1st `  F ) `
 P ) --> ( ( 1st `  G
) `  P )
)
9089adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( A `  P ) : ( ( 1st `  F
) `  P ) --> ( ( 1st `  G
) `  P )
)
9118, 56, 11, 67, 74, 80, 85, 90setcco 13931 . . . . . . . . . 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
) ) )
9255, 91eqtrd 2328 . . . . . . . . 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 ) ) )
9392oveq1d 5889 . . . . . . . 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
) ) )
9442, 29ffvelrnd 5682 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 1st `  Y
) `  P )  e.  ( O  Func  S
) )
95 1st2ndbr 6185 . . . . . . . . . . . . . 14  |-  ( ( Rel  ( O  Func  S )  /\  ( ( 1st `  Y ) `
 P )  e.  ( O  Func  S
) )  ->  ( 1st `  ( ( 1st `  Y ) `  P
) ) ( O 
Func  S ) ( 2nd `  ( ( 1st `  Y
) `  P )
) )
9658, 94, 95sylancr 644 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  P )
) ( O  Func  S ) ( 2nd `  (
( 1st `  Y
) `  P )
) )
9710, 57, 96funcf1 13756 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  P )
) : B --> ( Base `  S ) )
9897, 29ffvelrnd 5682 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  P )
) `  P )  e.  ( Base `  S
) )
9998, 63eleqtrrd 2373 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  P )
) `  P )  e.  U )
10099adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( 1st `  ( ( 1st `  Y
) `  P )
) `  P )  e.  U )
1017, 33nat1st2nd 13841 . . . . . . . . . . . 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 )
) >. ) )
1027, 101, 10, 82, 29natcl 13843 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
)  e.  ( ( ( 1st `  (
( 1st `  Y
) `  P )
) `  P )
(  Hom  `  S ) ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
) )
10318, 24, 82, 99, 66elsetchom 13929 . . . . . . . . . . 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 )
) )
104102, 103mpbid 201 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) : ( ( 1st `  ( ( 1st `  Y ) `
 P ) ) `
 P ) --> ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
)
105104adantr 451 . . . . . . . . 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 )
)
106 fco 5414 . . . . . . . . . 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 )
)
10790, 85, 106syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( A `
 P )  o.  ( a `  P
) ) : ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P ) --> ( ( 1st `  G
) `  P )
)
10818, 56, 11, 100, 67, 80, 105, 107setcco 13931 . . . . . . . 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 ) ) )
10954, 93, 1083eqtrd 2332 . . . . . . 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 )
) )
110109fveq1d 5543 . . . . . 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 )
) )
111 yoneda.1 . . . . . . . . . 10  |-  .1.  =  ( Id `  C )
1129, 13, 111, 17, 29catidcl 13600 . . . . . . . . 9  |-  ( ph  ->  (  .1.  `  P
)  e.  ( P (  Hom  `  C
) P ) )
11316, 9, 17, 29, 13, 29yon11 14054 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  P )
) `  P )  =  ( P (  Hom  `  C ) P ) )
114112, 113eleqtrrd 2373 . . . . . . . 8  |-  ( ph  ->  (  .1.  `  P
)  e.  ( ( 1st `  ( ( 1st `  Y ) `
 P ) ) `
 P ) )
115114adantr 451 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  (  .1.  `  P )  e.  ( ( 1st `  (
( 1st `  Y
) `  P )
) `  P )
)
116 fvco3 5612 . . . . . . 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 ) ) ) )
117105, 115, 116syl2anc 642 . . . . . 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 ) ) ) )
118105, 115ffvelrnd 5682 . . . . . . . 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 )
)
119 fvco3 5612 . . . . . . . 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 ) ) ) ) )
12085, 118, 119syl2anc 642 . . . . . . 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 ) ) ) ) )
12117adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  C  e.  Cat )
12230adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  X  e.  B
)
123 eqid 2296 . . . . . . . . . . . 12  |-  (comp `  C )  =  (comp `  C )
12432adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  K  e.  ( P (  Hom  `  C
) X ) )
125112adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  (  .1.  `  P )  e.  ( P (  Hom  `  C
) P ) )
12616, 9, 121, 53, 13, 122, 123, 53, 124, 125yon2 14056 . . . . . . . . . . 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
) ) )
1279, 13, 111, 121, 53, 123, 122, 124catrid 13602 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( K (
<. P ,  P >. (comp `  C ) X ) (  .1.  `  P
) )  =  K )
128126, 127eqtrd 2328 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( ( P ( 2nd `  Y ) X ) `
 K ) `  P ) `  (  .1.  `  P ) )  =  K )
129128fveq2d 5545 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( a `
 P ) `  ( ( ( ( P ( 2nd `  Y
) X ) `  K ) `  P
) `  (  .1.  `  P ) ) )  =  ( ( a `
 P ) `  K ) )
130 eqid 2296 . . . . . . . . . . . . . . 15  |-  (  Hom  `  O )  =  (  Hom  `  O )
13110, 130, 82, 60, 30, 29funcf2 13758 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( X ( 2nd `  ( ( 1st `  Y
) `  X )
) P ) : ( X (  Hom  `  O ) P ) --> ( ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  X )
(  Hom  `  S ) ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
) )
13213, 8oppchom 13634 . . . . . . . . . . . . . . 15  |-  ( X (  Hom  `  O
) P )  =  ( P (  Hom  `  C ) X )
13332, 132syl6eleqr 2387 . . . . . . . . . . . . . 14  |-  ( ph  ->  K  e.  ( X (  Hom  `  O
) P ) )
134131, 133ffvelrnd 5682 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( X ( 2nd `  ( ( 1st `  Y ) `
 X ) ) P ) `  K
)  e.  ( ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X )
(  Hom  `  S ) ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
) )
13565, 30ffvelrnd 5682 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X )  e.  U )
13618, 24, 82, 135, 66elsetchom 13929 . . . . . . . . . . . . 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 )
) )
137134, 136mpbid 201 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( X ( 2nd `  ( ( 1st `  Y ) `
 X ) ) P ) `  K
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 X ) --> ( ( 1st `  (
( 1st `  Y
) `  X )
) `  P )
)
138137adantr 451 . . . . . . . . . . 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 )
)
1399, 13, 111, 17, 30catidcl 13600 . . . . . . . . . . . . 13  |-  ( ph  ->  (  .1.  `  X
)  e.  ( X (  Hom  `  C
) X ) )
14016, 9, 17, 30, 13, 30yon11 14054 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X )  =  ( X (  Hom  `  C ) X ) )
141139, 140eleqtrrd 2373 . . . . . . . . . . . 12  |-  ( ph  ->  (  .1.  `  X
)  e.  ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 X ) )
142141adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  (  .1.  `  X )  e.  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X )
)
143 fvco3 5612 . . . . . . . . . . 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 ) ) ) )
144138, 142, 143syl2anc 642 . . . . . . . . . 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 ) ) ) )
145133adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  K  e.  ( X (  Hom  `  O
) P ) )
1467, 81, 10, 130, 11, 122, 53, 145nati 13845 . . . . . . . . . . . 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 ) ) )
147135adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  X )  e.  U )
14818, 56, 11, 147, 67, 74, 138, 85setcco 13931 . . . . . . . . . . . 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 ) ) )
14972, 30ffvelrnd 5682 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 1st `  F
) `  X )  e.  U )
150149adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( 1st `  F ) `  X
)  e.  U )
1517, 81, 10, 82, 122natcl 13843 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( a `  X )  e.  ( ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X )
(  Hom  `  S ) ( ( 1st `  F
) `  X )
) )
15218, 56, 82, 147, 150elsetchom 13929 . . . . . . . . . . . . . 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 )
) )
153151, 152mpbid 201 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( a `  X ) : ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X ) --> ( ( 1st `  F
) `  X )
)
15410, 130, 82, 69, 30, 29funcf2 13758 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( X ( 2nd `  F ) P ) : ( X (  Hom  `  O ) P ) --> ( ( ( 1st `  F
) `  X )
(  Hom  `  S ) ( ( 1st `  F
) `  P )
) )
155154, 133ffvelrnd 5682 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( X ( 2nd `  F ) P ) `  K
)  e.  ( ( ( 1st `  F
) `  X )
(  Hom  `  S ) ( ( 1st `  F
) `  P )
) )
15618, 24, 82, 149, 73elsetchom 13929 . . . . . . . . . . . . . . 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 )
) )
157155, 156mpbid 201 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( X ( 2nd `  F ) P ) `  K
) : ( ( 1st `  F ) `
 X ) --> ( ( 1st `  F
) `  P )
)
158157adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( X ( 2nd `  F
) P ) `  K ) : ( ( 1st `  F
) `  X ) --> ( ( 1st `  F
) `  P )
)
15918, 56, 11, 147, 150, 74, 153, 158setcco 13931 . . . . . . . . . . . 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 )
) )
160146, 148, 1593eqtr3d 2336 . . . . . . . . . . 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 ) ) )
161160fveq1d 5543 . . . . . . . . . 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 ) ) )
162139adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  (  .1.  `  X )  e.  ( X (  Hom  `  C
) X ) )
16316, 9, 121, 122, 13, 122, 123, 53, 124, 162yon12 14055 . . . . . . . . . . . 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 ) )
1649, 13, 111, 121, 53, 123, 122, 124catlid 13601 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( (  .1.  `  X ) ( <. P ,  X >. (comp `  C ) X ) K )  =  K )
165163, 164eqtrd 2328 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( X ( 2nd `  (
( 1st `  Y
) `  X )
) P ) `  K ) `  (  .1.  `  X ) )  =  K )
166165fveq2d 5545 . . . . . . . . . 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 ) )
167144, 161, 1663eqtr3d 2336 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( ( ( X ( 2nd `  F ) P ) `
 K )  o.  ( a `  X
) ) `  (  .1.  `  X ) )  =  ( ( a `
 P ) `  K ) )
168 fvco3 5612 . . . . . . . . . 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 ) ) ) )
169153, 142, 168syl2anc 642 . . . . . . . . 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 ) ) ) )
170129, 167, 1693eqtr2d 2334 . . . . . . . 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 ) ) ) )
171170fveq2d 5545 . . . . . . 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 ) ) ) ) )
172120, 171eqtrd 2328 . . . . . 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 ) ) ) ) )
173110, 117, 1723eqtrd 2332 . . . . 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 ) ) ) ) )
174173mpteq2dva 4122 . . . 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 ) ) ) ) ) )
1755, 174syl5eq 2340 . . 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 ) ) ) ) ) )
176 eqid 2296 . . . . . . . . . . . 12  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
177176, 35, 10xpcbas 13968 . . . . . . . . . . 11  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
178 eqid 2296 . . . . . . . . . . 11  |-  (  Hom  `  ( Q  X.c  O ) )  =  (  Hom  `  ( Q  X.c  O ) )
179 eqid 2296 . . . . . . . . . . 11  |-  (  Hom  `  T )  =  (  Hom  `  T )
180 relfunc 13752 . . . . . . . . . . . 12  |-  Rel  (
( Q  X.c  O ) 
Func  T )
181 yoneda.t . . . . . . . . . . . . . 14  |-  T  =  ( SetCat `  V )
182 yoneda.h . . . . . . . . . . . . . 14  |-  H  =  (HomF
`  Q )
183 yoneda.r . . . . . . . . . . . . . 14  |-  R  =  ( ( Q  X.c  O
) FuncCat  T )
184 yoneda.e . . . . . . . . . . . . . 14  |-  E  =  ( O evalF  S )
185 yoneda.z . . . . . . . . . . . . . 14  |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )
18616, 9, 111, 8, 18, 181, 6, 182, 183, 184, 185, 17, 22, 25, 20yonedalem1 14062 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Z  e.  ( ( Q  X.c  O ) 
Func  T )  /\  E  e.  ( ( Q  X.c  O
)  Func  T )
) )
187186simpld 445 . . . . . . . . . . . 12  |-  ( ph  ->  Z  e.  ( ( Q  X.c  O )  Func  T
) )
188 1st2ndbr 6185 . . . . . . . . . . . 12  |-  ( ( 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
) )
189180, 187, 188sylancr 644 . . . . . . . . . . 11  |-  ( ph  ->  ( 1st `  Z
) ( ( Q  X.c  O )  Func  T
) ( 2nd `  Z
) )
190 opelxpi 4737 . . . . . . . . . . . 12  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  <. F ,  X >.  e.  ( ( O  Func  S )  X.  B ) )
19145, 30, 190syl2anc 642 . . . . . . . . . . 11  |-  ( ph  -> 
<. F ,  X >.  e.  ( ( O  Func  S )  X.  B ) )
192 opelxpi 4737 . . . . . . . . . . . 12  |-  ( ( G  e.  ( O 
Func  S )  /\  P  e.  B )  ->  <. G ,  P >.  e.  ( ( O  Func  S )  X.  B ) )
19347, 29, 192syl2anc 642 . . . . . . . . . . 11  |-  ( ph  -> 
<. G ,  P >.  e.  ( ( O  Func  S )  X.  B ) )
194177, 178, 179, 189, 191, 193funcf2 13758 . . . . . . . . . 10  |-  ( 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 >. ) ) )
195176, 35, 10, 14, 130, 45, 30, 47, 29, 178xpchom2 13976 . . . . . . . . . . . 12  |-  ( ph  ->  ( <. F ,  X >. (  Hom  `  ( Q  X.c  O ) ) <. G ,  P >. )  =  ( ( F ( O Nat  S ) G )  X.  ( X (  Hom  `  O
) P ) ) )
196132xpeq2i 4726 . . . . . . . . . . . 12  |-  ( ( F ( O Nat  S
) G )  X.  ( X (  Hom  `  O ) P ) )  =  ( ( F ( O Nat  S
) G )  X.  ( P (  Hom  `  C ) X ) )
197195, 196syl6eq 2344 . . . . . . . . . . 11  |-  ( ph  ->  ( <. F ,  X >. (  Hom  `  ( Q  X.c  O ) ) <. G ,  P >. )  =  ( ( F ( O Nat  S ) G )  X.  ( P (  Hom  `  C
) X ) ) )
198 df-ov 5877 . . . . . . . . . . . . . 14  |-  ( F ( 1st `  Z
) X )  =  ( ( 1st `  Z
) `  <. F ,  X >. )
199 df-ov 5877 . . . . . . . . . . . . . 14  |-  ( G ( 1st `  Z
) P )  =  ( ( 1st `  Z
) `  <. G ,  P >. )
200198, 199oveq12i 5886 . . . . . . . . . . . . 13  |-  ( ( F ( 1st `  Z
) X ) (  Hom  `  T )
( G ( 1st `  Z ) P ) )  =  ( ( ( 1st `  Z
) `  <. F ,  X >. ) (  Hom  `  T ) ( ( 1st `  Z ) `
 <. G ,  P >. ) )
201200eqcomi 2300 . . . . . . . . . . . 12  |-  ( ( ( 1st `  Z
) `  <. F ,  X >. ) (  Hom  `  T ) ( ( 1st `  Z ) `
 <. G ,  P >. ) )  =  ( ( F ( 1st `  Z ) X ) (  Hom  `  T
) ( G ( 1st `  Z ) P ) )
202201a1i 10 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 1st `  Z ) `  <. F ,  X >. )
(  Hom  `  T ) ( ( 1st `  Z
) `  <. G ,  P >. ) )  =  ( ( F ( 1st `  Z ) X ) (  Hom  `  T ) ( G ( 1st `  Z
) P ) ) )
203197, 202feq23d 5402 . . . . . . . . . 10  |-  ( 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 ) ) ) )
204194, 203mpbid 201 . . . . . . . . 9  |-  ( 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 ) ) )
205204, 50, 32fovrnd 6008 . . . . . . . 8  |-  ( ph  ->  ( A ( <. F ,  X >. ( 2nd `  Z )
<. G ,  P >. ) K )  e.  ( ( F ( 1st `  Z ) X ) (  Hom  `  T
) ( G ( 1st `  Z ) P ) ) )
206 eqid 2296 . . . . . . . . . . . 12  |-  ( Base `  T )  =  (
Base `  T )
207177, 206, 189funcf1 13756 . . . . . . . . . . 11  |-  ( ph  ->  ( 1st `  Z
) : ( ( O  Func  S )  X.  B ) --> ( Base `  T ) )
208207, 45, 30fovrnd 6008 . . . . . . . . . 10  |-  ( ph  ->  ( F ( 1st `  Z ) X )  e.  ( Base `  T
) )
209181, 22setcbas 13926 . . . . . . . . . 10  |-  ( ph  ->  V  =  ( Base `  T ) )
210208, 209eleqtrrd 2373 . . . . . . . . 9  |-  ( ph  ->  ( F ( 1st `  Z ) X )  e.  V )
211207, 47, 29fovrnd 6008 . . . . . . . . . 10  |-  ( ph  ->  ( G ( 1st `  Z ) P )  e.  ( Base `  T
) )
212211, 209eleqtrrd 2373 . . . . . . . . 9  |-  ( ph  ->  ( G ( 1st `  Z ) P )  e.  V )
213181, 22, 179, 210, 212elsetchom 13929 . . . . . . . 8  |-  ( 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 ) ) )
214205, 213mpbid 201 . . . . . . 7  |-  ( ph  ->  ( A ( <. F ,  X >. ( 2nd `  Z )
<. G ,  P >. ) K ) : ( F ( 1st `  Z
) X ) --> ( G ( 1st `  Z
) P ) )
21516, 9, 111, 8, 18, 181, 6, 182, 183, 184, 185, 17, 22, 25, 20, 45, 30, 47, 29, 50, 32yonedalem22 14068 . . . . . . . . 9  |-  ( 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 ) )
216182, 40, 35, 14, 43, 45, 94, 47, 12, 33, 50hof2val 14046 . . . . . . . . 9  |-  ( 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 ) ) ) )
217215, 216eqtrd 2328 . . . . . . . 8  |-  ( 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 ) ) ) )
21816, 9, 111, 8, 18, 181, 6, 182, 183, 184, 185, 17, 22, 25, 20, 45, 30yonedalem21 14063 . . . . . . . 8  |-  ( ph  ->  ( F ( 1st `  Z ) X )  =  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F ) )
21916, 9, 111, 8, 18, 181, 6, 182, 183, 184, 185, 17, 22, 25, 20, 47, 29yonedalem21 14063 . . . . . . . 8  |-  ( ph  ->  ( G ( 1st `  Z ) P )  =  ( ( ( 1st `  Y ) `
 P ) ( O Nat  S ) G ) )
220217, 218, 219feq123d 5398 . . . . . . 7  |-  ( 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 ) ) )
221214, 220mpbid 201 . . . . . 6  |-  ( 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 ) )
222 eqid 2296 . . . . . . 7  |-  ( 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 ) ) )
223222fmpt 5697 . . . . . 6  |-  ( 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 ) )
224221, 223sylibr 203 . . . . 5  |-  ( 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 ) )
225224r19.21bi 2654 . . . 4  |-  ( (
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 ) )  e.  ( ( ( 1st `  Y ) `
 P ) ( O Nat  S ) G ) )
226 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 ) ) ) )
22716, 9, 111, 8, 18, 181, 6, 182, 183, 184, 185, 17, 22, 25, 20, 47, 29, 226yonedalem3a 14064 . . . . 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 ) ) )
228227simpld 445 . . . 4  |-  ( ph  ->  ( G M P )  =  ( a  e.  ( ( ( 1st `  Y ) `
 P ) ( O Nat  S ) G )  |->  ( ( a `
 P ) `  (  .1.  `  P )
) ) )
229 fveq1 5540 . . . . 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 ) )
230229fveq1d 5543 . . . 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 )
) )
231225, 217, 228, 230fmptco 5707 . . 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 )
) ) )
232 eqid 2296 . . . . . . 7  |-  ( <. F ,  X >. ( 2nd `  E )
<. G ,  P >. )  =  ( <. F ,  X >. ( 2nd `  E
) <. G ,  P >. )
233184, 37, 39, 10, 130, 11, 7, 45, 47, 30, 29, 232, 50, 133evlf2val 14009 . . . . . 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 ) ) )
23418, 24, 11, 149, 73, 79, 157, 89setcco 13931 . . . . . 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 ) ) )
235233, 234eqtrd 2328 . . . . 5  |-  ( ph  ->  ( A ( <. F ,  X >. ( 2nd `  E )
<. G ,  P >. ) K )  =  ( ( A `  P
)  o.  ( ( X ( 2nd `  F
) P ) `  K ) ) )
236235coeq1d 4861 . . . 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 ) ) )
23716, 9, 111, 8, 18, 181, 6, 182, 183, 184, 185, 17, 22, 25, 20, 45, 30, 226yonedalem3a 14064 . . . . . . . . 9  |-  ( 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 ) ) )
238237simprd 449 . . . . . . . 8  |-  ( ph  ->  ( F M X ) : ( F ( 1st `  Z
) X ) --> ( F ( 1st `  E
) X ) )
239237simpld 445 . . . . . . . . 9  |-  ( ph  ->  ( F M X )  =  ( a  e.  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F )  |->  ( ( a `
 X ) `  (  .1.  `  X )
) ) )
240184, 37, 39, 10, 45, 30evlf1 14010 . . . . . . . . 9  |-  ( ph  ->  ( F ( 1st `  E ) X )  =  ( ( 1st `  F ) `  X
) )
241239, 218, 240feq123d 5398 . . . . . . . 8  |-  ( 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
) ) )
242238, 241mpbid 201 . . . . . . 7  |-  ( ph  ->  ( a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  |->  ( ( a `  X ) `
 (  .1.  `  X ) ) ) : ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F ) --> ( ( 1st `  F ) `  X
) )
243 eqid 2296 . . . . . . . 8  |-  ( 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 ) ) )
244243fmpt 5697 . . . . . . 7  |-  ( 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
) )
245242, 244sylibr 203 . . . . . 6  |-  ( ph  ->  A. a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F ) ( ( a `  X ) `
 (  .1.  `  X ) )  e.  ( ( 1st `  F
) `  X )
)
246245r19.21bi 2654 . . . . 5  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( a `
 X ) `  (  .1.  `  X )
)  e.  ( ( 1st `  F ) `
 X ) )
247 fcompt 5710 . . . . . 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 ) ) ) )
24889, 157, 247syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( A `  P )  o.  (
( X ( 2nd `  F ) P ) `
 K ) )  =  ( y  e.  ( ( 1st `  F
) `  X )  |->  ( ( A `  P ) `  (
( ( X ( 2nd `  F ) P ) `  K
) `  y )
) ) )
249 fveq2 5541 . . . . . 6  |-  ( y  =  ( ( a `
 X ) `  (  .1.  `  X )
)  ->  ( (
( X ( 2nd `  F ) P ) `
 K ) `  y )  =  ( ( ( X ( 2nd `  F ) P ) `  K
) `  ( (
a `  X ) `  (  .1.  `  X
) ) ) )
250249fveq2d 5545 . . . . 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 ) ) ) ) )
251246, 239, 248, 250fmptco 5707 . . . 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 ) ) ) ) ) )
252236, 251eqtrd 2328 . . 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 ) ) ) ) ) )
253175, 231, 2523eqtr4d 2338 . 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 ) ) )
254 eqid 2296 . . 3  |-  (comp `  T )  =  (comp `  T )
255186simprd 449 . . . . . . 7  |-  ( ph  ->  E  e.  ( ( Q  X.c  O )  Func  T
) )
256 1st2ndbr 6185 . . . . . . 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
) )
257180, 255, 256sylancr 644 . . . . . 6  |-  ( ph  ->  ( 1st `  E
) ( ( Q  X.c  O )  Func  T
) ( 2nd `  E
) )
258177, 206, 257funcf1 13756 . . . . 5  |-  ( ph  ->  ( 1st `  E
) : ( ( O  Func  S )  X.  B ) --> ( Base `  T ) )
259258, 47, 29fovrnd 6008 . . . 4  |-  ( ph  ->  ( G ( 1st `  E ) P )  e.  ( Base `  T
) )
260259, 209eleqtrrd 2373 . . 3  |-  ( ph  ->  ( G ( 1st `  E ) P )  e.  V )
261227simprd 449 . . 3  |-  ( ph  ->  ( G M P ) : ( G ( 1st `  Z
) P ) --> ( G ( 1st `  E
) P ) )
262181, 22, 254, 210, 212, 260, 214, 261setcco 13931 . 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 ) ) )
263258, 45, 30fovrnd 6008 . . . 4  |-  ( ph  ->  ( F ( 1st `  E ) X )  e.  ( Base `  T
) )
264263, 209eleqtrrd 2373 . . 3  |-  ( ph  ->  ( F ( 1st `  E ) X )  e.  V )
265177, 178, 179, 257, 191, 193funcf2 13758 . . . . . 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 >. ) ) )
266 df-ov 5877 . . . . . . . . . 10  |-  ( F ( 1st `  E
) X )  =  ( ( 1st `  E
) `  <. F ,  X >. )
267 df-ov 5877 . . . . . . . . . 10  |-  ( G ( 1st `  E
) P )  =  ( ( 1st `  E
) `  <. G ,  P >. )
268266, 267oveq12i 5886 . . . . . . . . 9  |-  ( ( F ( 1st `  E
) X ) (  Hom  `  T )
( G ( 1st `  E ) P ) )  =  ( ( ( 1st `  E
) `  <. F ,  X >. ) (  Hom  `  T ) ( ( 1st `  E ) `
 <. G ,  P >. ) )
269268eqcomi 2300 . . . . . . . 8  |-  ( ( ( 1st `  E
) `  <. F ,  X >. ) (  Hom  `  T ) ( ( 1st `  E ) `
 <. G ,  P >. ) )  =  ( ( F ( 1st `  E ) X ) (  Hom  `  T
) ( G ( 1st `  E ) P ) )
270269a1i 10 . . . . . . 7  |-  ( ph  ->  ( ( ( 1st `  E ) `  <. F ,  X >. )
(  Hom  `  T ) ( ( 1st `  E
) `  <. G ,  P >. ) )  =  ( ( F ( 1st `  E ) X ) (  Hom  `  T ) ( G ( 1st `  E
) P ) ) )
271197, 270feq23d 5402 . . . . . 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 ) ) ) )
272265, 271mpbid 201 . . . . 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 ) ) )
273272, 50, 32fovrnd 6008 . . . 4  |-  ( ph  ->  ( A ( <. F ,  X >. ( 2nd `  E )
<. G ,  P >. ) K )  e.  ( ( F ( 1st `  E ) X ) (  Hom  `  T
) ( G ( 1st `  E ) P ) ) )
274181, 22, 179, 264, 260elsetchom 13929 . . . 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 ) ) )
275273, 274mpbid 201 . . 3  |-  ( ph  ->  ( A ( <. F ,  X >. ( 2nd `  E )
<. G ,  P >. ) K ) : ( F ( 1st `  E
) X ) --> ( G ( 1st `  E
) P ) )
276181, 22, 254, 210, 264, 260, 238, 275setcco 13931 . 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 ) ) )
277253, 262, 2763eqtr4d 2338 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 set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    u. cun 3163    C_ wss 3165   <.cop 3656   class class class wbr 4039    e. cmpt 4093    X. cxp 4703   ran crn 4706    o. ccom 4709   Rel wrel 4710   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137  tpos ctpos 6249   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582   Idccid 13583    Homf chomf 13584  oppCatcoppc 13630    Func cfunc 13744    o.func ccofu 13746   Nat cnat 13831   FuncCat cfuc 13832   SetCatcsetc 13923    X.c cxpc 13958    1stF c1stf 13959    2ndF c2ndf 13960   ⟨,⟩F cprf 13961   evalF cevlf 13999  HomFchof 14038  Yoncyon 14039
This theorem is referenced by:  yonedalem3  14070
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-hom 13248  df-cco 13249  df-cat 13586  df-cid 13587  df-homf 13588  df-comf 13589  df-oppc 13631  df-ssc 13703  df-resc 13704  df-subc 13705  df-func 13748  df-cofu 13750  df-nat 13833  df-fuc 13834  df-setc 13924  df-xpc 13962  df-1stf 13963  df-2ndf 13964  df-prf 13965  df-evlf 14003  df-curf 14004  df-hof 14040  df-yon 14041
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