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Theorem yonedalem4c 15400
Description: Lemma for yoneda 15406. (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 )
yonedalem4.n  |-  N  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( u  e.  ( ( 1st `  f
) `  x )  |->  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C
) x )  |->  ( ( ( x ( 2nd `  f ) y ) `  g
) `  u )
) ) ) )
yonedalem4.p  |-  ( ph  ->  A  e.  ( ( 1st `  F ) `
 X ) )
Assertion
Ref Expression
yonedalem4c  |-  ( ph  ->  ( ( F N X ) `  A
)  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F ) )
Distinct variable groups:    f, g, x, y,  .1.    u, g, A, y    u, f, C, g, x, y   
f, E, g, u, y    f, F, g, u, x, y    B, f, g, u, x, y   
f, O, g, u, x, y    S, f, g, u, x, y    Q, f, g, u, x    T, f, g, u, y    ph, f, g, u, x, y    u, R    f, Y, g, u, x, y   
f, Z, g, u, x, y    f, X, g, u, x, y
Allowed substitution hints:    A( x, f)    Q( y)    R( x, y, f, g)    T( x)    U( x, y, u, f, g)    .1. ( u)    E( x)    H( x, y, u, f, g)    N( x, y, u, f, g)    V( x, y, u, f, g)    W( x, y, u, f, g)

Proof of Theorem yonedalem4c
Dummy variables  h  k  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 yoneda.y . . . . 5  |-  Y  =  (Yon `  C )
2 yoneda.b . . . . 5  |-  B  =  ( Base `  C
)
3 yoneda.1 . . . . 5  |-  .1.  =  ( Id `  C )
4 yoneda.o . . . . 5  |-  O  =  (oppCat `  C )
5 yoneda.s . . . . 5  |-  S  =  ( SetCat `  U )
6 yoneda.t . . . . 5  |-  T  =  ( SetCat `  V )
7 yoneda.q . . . . 5  |-  Q  =  ( O FuncCat  S )
8 yoneda.h . . . . 5  |-  H  =  (HomF
`  Q )
9 yoneda.r . . . . 5  |-  R  =  ( ( Q  X.c  O
) FuncCat  T )
10 yoneda.e . . . . 5  |-  E  =  ( O evalF  S )
11 yoneda.z . . . . 5  |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )
12 yoneda.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
13 yoneda.w . . . . 5  |-  ( ph  ->  V  e.  W )
14 yoneda.u . . . . 5  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
15 yoneda.v . . . . 5  |-  ( ph  ->  ( ran  ( Hom f  `  Q )  u.  U
)  C_  V )
16 yonedalem21.f . . . . 5  |-  ( ph  ->  F  e.  ( O 
Func  S ) )
17 yonedalem21.x . . . . 5  |-  ( ph  ->  X  e.  B )
18 yonedalem4.n . . . . 5  |-  N  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( u  e.  ( ( 1st `  f
) `  x )  |->  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C
) x )  |->  ( ( ( x ( 2nd `  f ) y ) `  g
) `  u )
) ) ) )
19 yonedalem4.p . . . . 5  |-  ( ph  ->  A  e.  ( ( 1st `  F ) `
 X ) )
201, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19yonedalem4a 15398 . . . 4  |-  ( ph  ->  ( ( F N X ) `  A
)  =  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A ) ) ) )
21 oveq1 6289 . . . . . 6  |-  ( y  =  z  ->  (
y ( Hom  `  C
) X )  =  ( z ( Hom  `  C ) X ) )
22 oveq2 6290 . . . . . . . 8  |-  ( y  =  z  ->  ( X ( 2nd `  F
) y )  =  ( X ( 2nd `  F ) z ) )
2322fveq1d 5866 . . . . . . 7  |-  ( y  =  z  ->  (
( X ( 2nd `  F ) y ) `
 g )  =  ( ( X ( 2nd `  F ) z ) `  g
) )
2423fveq1d 5866 . . . . . 6  |-  ( y  =  z  ->  (
( ( X ( 2nd `  F ) y ) `  g
) `  A )  =  ( ( ( X ( 2nd `  F
) z ) `  g ) `  A
) )
2521, 24mpteq12dv 4525 . . . . 5  |-  ( y  =  z  ->  (
g  e.  ( y ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) y ) `  g
) `  A )
)  =  ( g  e.  ( z ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) z ) `
 g ) `  A ) ) )
2625cbvmptv 4538 . . . 4  |-  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A ) ) )  =  ( z  e.  B  |->  ( g  e.  ( z ( Hom  `  C ) X ) 
|->  ( ( ( X ( 2nd `  F
) z ) `  g ) `  A
) ) )
2720, 26syl6eq 2524 . . 3  |-  ( ph  ->  ( ( F N X ) `  A
)  =  ( z  e.  B  |->  ( g  e.  ( z ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) z ) `
 g ) `  A ) ) ) )
284, 2oppcbas 14970 . . . . . . . . . . . . 13  |-  B  =  ( Base `  O
)
29 eqid 2467 . . . . . . . . . . . . 13  |-  ( Hom  `  O )  =  ( Hom  `  O )
30 eqid 2467 . . . . . . . . . . . . 13  |-  ( Hom  `  S )  =  ( Hom  `  S )
31 relfunc 15085 . . . . . . . . . . . . . . 15  |-  Rel  ( O  Func  S )
32 1st2ndbr 6830 . . . . . . . . . . . . . . 15  |-  ( ( Rel  ( O  Func  S )  /\  F  e.  ( O  Func  S
) )  ->  ( 1st `  F ) ( O  Func  S )
( 2nd `  F
) )
3331, 16, 32sylancr 663 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1st `  F
) ( O  Func  S ) ( 2nd `  F
) )
3433adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  B )  ->  ( 1st `  F ) ( O  Func  S )
( 2nd `  F
) )
3517adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  B )  ->  X  e.  B )
36 simpr 461 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  B )  ->  z  e.  B )
3728, 29, 30, 34, 35, 36funcf2 15091 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  B )  ->  ( X ( 2nd `  F
) z ) : ( X ( Hom  `  O ) z ) --> ( ( ( 1st `  F ) `  X
) ( Hom  `  S
) ( ( 1st `  F ) `  z
) ) )
3837adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z ( Hom  `  C ) X ) )  ->  ( X
( 2nd `  F
) z ) : ( X ( Hom  `  O ) z ) --> ( ( ( 1st `  F ) `  X
) ( Hom  `  S
) ( ( 1st `  F ) `  z
) ) )
39 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z ( Hom  `  C ) X ) )  ->  g  e.  ( z ( Hom  `  C ) X ) )
40 eqid 2467 . . . . . . . . . . . . 13  |-  ( Hom  `  C )  =  ( Hom  `  C )
4140, 4oppchom 14967 . . . . . . . . . . . 12  |-  ( X ( Hom  `  O
) z )  =  ( z ( Hom  `  C ) X )
4239, 41syl6eleqr 2566 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z ( Hom  `  C ) X ) )  ->  g  e.  ( X ( Hom  `  O
) z ) )
4338, 42ffvelrnd 6020 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z ( Hom  `  C ) X ) )  ->  ( ( X ( 2nd `  F
) z ) `  g )  e.  ( ( ( 1st `  F
) `  X )
( Hom  `  S ) ( ( 1st `  F
) `  z )
) )
4415unssbd 3682 . . . . . . . . . . . . . 14  |-  ( ph  ->  U  C_  V )
4513, 44ssexd 4594 . . . . . . . . . . . . 13  |-  ( ph  ->  U  e.  _V )
4645adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  B )  ->  U  e.  _V )
4746adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z ( Hom  `  C ) X ) )  ->  U  e.  _V )
48 eqid 2467 . . . . . . . . . . . . . . 15  |-  ( Base `  S )  =  (
Base `  S )
4928, 48, 33funcf1 15089 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  S ) )
50 eqidd 2468 . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  =  B )
515, 45setcbas 15259 . . . . . . . . . . . . . . 15  |-  ( ph  ->  U  =  ( Base `  S ) )
5250, 51feq23d 5724 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 1st `  F
) : B --> U  <->  ( 1st `  F ) : B --> ( Base `  S )
) )
5349, 52mpbird 232 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1st `  F
) : B --> U )
5453, 17ffvelrnd 6020 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1st `  F
) `  X )  e.  U )
5554ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z ( Hom  `  C ) X ) )  ->  ( ( 1st `  F ) `  X )  e.  U
)
5653ffvelrnda 6019 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  F
) `  z )  e.  U )
5756adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z ( Hom  `  C ) X ) )  ->  ( ( 1st `  F ) `  z )  e.  U
)
585, 47, 30, 55, 57elsetchom 15262 . . . . . . . . . 10  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z ( Hom  `  C ) X ) )  ->  ( (
( X ( 2nd `  F ) z ) `
 g )  e.  ( ( ( 1st `  F ) `  X
) ( Hom  `  S
) ( ( 1st `  F ) `  z
) )  <->  ( ( X ( 2nd `  F
) z ) `  g ) : ( ( 1st `  F
) `  X ) --> ( ( 1st `  F
) `  z )
) )
5943, 58mpbid 210 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z ( Hom  `  C ) X ) )  ->  ( ( X ( 2nd `  F
) z ) `  g ) : ( ( 1st `  F
) `  X ) --> ( ( 1st `  F
) `  z )
)
6019ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z ( Hom  `  C ) X ) )  ->  A  e.  ( ( 1st `  F
) `  X )
)
6159, 60ffvelrnd 6020 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z ( Hom  `  C ) X ) )  ->  ( (
( X ( 2nd `  F ) z ) `
 g ) `  A )  e.  ( ( 1st `  F
) `  z )
)
62 eqid 2467 . . . . . . . 8  |-  ( g  e.  ( z ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) z ) `
 g ) `  A ) )  =  ( g  e.  ( z ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) z ) `  g
) `  A )
)
6361, 62fmptd 6043 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
g  e.  ( z ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) z ) `  g
) `  A )
) : ( z ( Hom  `  C
) X ) --> ( ( 1st `  F
) `  z )
)
6412adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  C  e.  Cat )
651, 2, 64, 35, 40, 36yon11 15387 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  Y
) `  X )
) `  z )  =  ( z ( Hom  `  C ) X ) )
6665feq2d 5716 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
( g  e.  ( z ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) z ) `  g
) `  A )
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 z ) --> ( ( 1st `  F
) `  z )  <->  ( g  e.  ( z ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) z ) `  g
) `  A )
) : ( z ( Hom  `  C
) X ) --> ( ( 1st `  F
) `  z )
) )
6763, 66mpbird 232 . . . . . 6  |-  ( (
ph  /\  z  e.  B )  ->  (
g  e.  ( z ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) z ) `  g
) `  A )
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 z ) --> ( ( 1st `  F
) `  z )
)
681, 2, 12, 17, 4, 5, 45, 14yon1cl 15386 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 1st `  Y
) `  X )  e.  ( O  Func  S
) )
69 1st2ndbr 6830 . . . . . . . . . . 11  |-  ( ( Rel  ( O  Func  S )  /\  ( ( 1st `  Y ) `
 X )  e.  ( O  Func  S
) )  ->  ( 1st `  ( ( 1st `  Y ) `  X
) ) ( O 
Func  S ) ( 2nd `  ( ( 1st `  Y
) `  X )
) )
7031, 68, 69sylancr 663 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
) ( O  Func  S ) ( 2nd `  (
( 1st `  Y
) `  X )
) )
7128, 48, 70funcf1 15089 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
) : B --> ( Base `  S ) )
7250, 51feq23d 5724 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) : B --> U  <->  ( 1st `  ( ( 1st `  Y
) `  X )
) : B --> ( Base `  S ) ) )
7371, 72mpbird 232 . . . . . . . 8  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
) : B --> U )
7473ffvelrnda 6019 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  Y
) `  X )
) `  z )  e.  U )
755, 46, 30, 74, 56elsetchom 15262 . . . . . 6  |-  ( (
ph  /\  z  e.  B )  ->  (
( g  e.  ( z ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) z ) `  g
) `  A )
)  e.  ( ( ( 1st `  (
( 1st `  Y
) `  X )
) `  z )
( Hom  `  S ) ( ( 1st `  F
) `  z )
)  <->  ( g  e.  ( z ( Hom  `  C ) X ) 
|->  ( ( ( X ( 2nd `  F
) z ) `  g ) `  A
) ) : ( ( 1st `  (
( 1st `  Y
) `  X )
) `  z ) --> ( ( 1st `  F
) `  z )
) )
7667, 75mpbird 232 . . . . 5  |-  ( (
ph  /\  z  e.  B )  ->  (
g  e.  ( z ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) z ) `  g
) `  A )
)  e.  ( ( ( 1st `  (
( 1st `  Y
) `  X )
) `  z )
( Hom  `  S ) ( ( 1st `  F
) `  z )
) )
7776ralrimiva 2878 . . . 4  |-  ( ph  ->  A. z  e.  B  ( g  e.  ( z ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) z ) `  g
) `  A )
)  e.  ( ( ( 1st `  (
( 1st `  Y
) `  X )
) `  z )
( Hom  `  S ) ( ( 1st `  F
) `  z )
) )
78 fvex 5874 . . . . . 6  |-  ( Base `  C )  e.  _V
792, 78eqeltri 2551 . . . . 5  |-  B  e. 
_V
80 mptelixpg 7503 . . . . 5  |-  ( B  e.  _V  ->  (
( z  e.  B  |->  ( g  e.  ( z ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) z ) `  g
) `  A )
) )  e.  X_ z  e.  B  (
( ( 1st `  (
( 1st `  Y
) `  X )
) `  z )
( Hom  `  S ) ( ( 1st `  F
) `  z )
)  <->  A. z  e.  B  ( g  e.  ( z ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) z ) `  g
) `  A )
)  e.  ( ( ( 1st `  (
( 1st `  Y
) `  X )
) `  z )
( Hom  `  S ) ( ( 1st `  F
) `  z )
) ) )
8179, 80ax-mp 5 . . . 4  |-  ( ( z  e.  B  |->  ( g  e.  ( z ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) z ) `  g
) `  A )
) )  e.  X_ z  e.  B  (
( ( 1st `  (
( 1st `  Y
) `  X )
) `  z )
( Hom  `  S ) ( ( 1st `  F
) `  z )
)  <->  A. z  e.  B  ( g  e.  ( z ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) z ) `  g
) `  A )
)  e.  ( ( ( 1st `  (
( 1st `  Y
) `  X )
) `  z )
( Hom  `  S ) ( ( 1st `  F
) `  z )
) )
8277, 81sylibr 212 . . 3  |-  ( ph  ->  ( z  e.  B  |->  ( g  e.  ( z ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) z ) `  g
) `  A )
) )  e.  X_ z  e.  B  (
( ( 1st `  (
( 1st `  Y
) `  X )
) `  z )
( Hom  `  S ) ( ( 1st `  F
) `  z )
) )
8327, 82eqeltrd 2555 . 2  |-  ( ph  ->  ( ( F N X ) `  A
)  e.  X_ z  e.  B  ( (
( 1st `  (
( 1st `  Y
) `  X )
) `  z )
( Hom  `  S ) ( ( 1st `  F
) `  z )
) )
8412adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  C  e.  Cat )
8517adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  X  e.  B )
86 simpr1 1002 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  z  e.  B )
871, 2, 84, 85, 40, 86yon11 15387 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( 1st `  (
( 1st `  Y
) `  X )
) `  z )  =  ( z ( Hom  `  C ) X ) )
8887eleq2d 2537 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
k  e.  ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 z )  <->  k  e.  ( z ( Hom  `  C ) X ) ) )
8988biimpa 484 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  z )
)  ->  k  e.  ( z ( Hom  `  C ) X ) )
90 eqid 2467 . . . . . . . . . . . 12  |-  (comp `  O )  =  (comp `  O )
91 eqid 2467 . . . . . . . . . . . 12  |-  (comp `  S )  =  (comp `  S )
9233adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  ( 1st `  F ) ( O  Func  S )
( 2nd `  F
) )
9392adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( 1st `  F
) ( O  Func  S ) ( 2nd `  F
) )
9485adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  X  e.  B
)
9586adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  z  e.  B
)
96 simpr2 1003 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  w  e.  B )
9796adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  w  e.  B
)
98 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  k  e.  ( z ( Hom  `  C
) X ) )
9998, 41syl6eleqr 2566 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  k  e.  ( X ( Hom  `  O
) z ) )
100 simplr3 1040 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  h  e.  ( z ( Hom  `  O
) w ) )
10128, 29, 90, 91, 93, 94, 95, 97, 99, 100funcco 15094 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( X ( 2nd `  F
) w ) `  ( h ( <. X ,  z >. (comp `  O ) w ) k ) )  =  ( ( ( z ( 2nd `  F
) w ) `  h ) ( <.
( ( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  z
) >. (comp `  S
) ( ( 1st `  F ) `  w
) ) ( ( X ( 2nd `  F
) z ) `  k ) ) )
102 eqid 2467 . . . . . . . . . . . . 13  |-  (comp `  C )  =  (comp `  C )
1032, 102, 4, 94, 95, 97oppcco 14969 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( h (
<. X ,  z >.
(comp `  O )
w ) k )  =  ( k (
<. w ,  z >.
(comp `  C ) X ) h ) )
104103fveq2d 5868 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( X ( 2nd `  F
) w ) `  ( h ( <. X ,  z >. (comp `  O ) w ) k ) )  =  ( ( X ( 2nd `  F ) w ) `  (
k ( <. w ,  z >. (comp `  C ) X ) h ) ) )
10545adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  U  e.  _V )
106105adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  U  e.  _V )
10754ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( 1st `  F ) `  X
)  e.  U )
108563ad2antr1 1161 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( 1st `  F
) `  z )  e.  U )
109108adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( 1st `  F ) `  z
)  e.  U )
11053adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  ( 1st `  F ) : B --> U )
111110, 96ffvelrnd 6020 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( 1st `  F
) `  w )  e.  U )
112111adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( 1st `  F ) `  w
)  e.  U )
11328, 29, 30, 92, 85, 86funcf2 15091 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  ( X ( 2nd `  F
) z ) : ( X ( Hom  `  O ) z ) --> ( ( ( 1st `  F ) `  X
) ( Hom  `  S
) ( ( 1st `  F ) `  z
) ) )
114113adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( X ( 2nd `  F ) z ) : ( X ( Hom  `  O
) z ) --> ( ( ( 1st `  F
) `  X )
( Hom  `  S ) ( ( 1st `  F
) `  z )
) )
115114, 99ffvelrnd 6020 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( X ( 2nd `  F
) z ) `  k )  e.  ( ( ( 1st `  F
) `  X )
( Hom  `  S ) ( ( 1st `  F
) `  z )
) )
1165, 106, 30, 107, 109elsetchom 15262 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( ( X ( 2nd `  F
) z ) `  k )  e.  ( ( ( 1st `  F
) `  X )
( Hom  `  S ) ( ( 1st `  F
) `  z )
)  <->  ( ( X ( 2nd `  F
) z ) `  k ) : ( ( 1st `  F
) `  X ) --> ( ( 1st `  F
) `  z )
) )
117115, 116mpbid 210 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( X ( 2nd `  F
) z ) `  k ) : ( ( 1st `  F
) `  X ) --> ( ( 1st `  F
) `  z )
)
11828, 29, 30, 92, 86, 96funcf2 15091 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
z ( 2nd `  F
) w ) : ( z ( Hom  `  O ) w ) --> ( ( ( 1st `  F ) `  z
) ( Hom  `  S
) ( ( 1st `  F ) `  w
) ) )
119 simpr3 1004 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  h  e.  ( z ( Hom  `  O ) w ) )
120118, 119ffvelrnd 6020 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( z ( 2nd `  F ) w ) `
 h )  e.  ( ( ( 1st `  F ) `  z
) ( Hom  `  S
) ( ( 1st `  F ) `  w
) ) )
1215, 105, 30, 108, 111elsetchom 15262 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( ( z ( 2nd `  F ) w ) `  h
)  e.  ( ( ( 1st `  F
) `  z )
( Hom  `  S ) ( ( 1st `  F
) `  w )
)  <->  ( ( z ( 2nd `  F
) w ) `  h ) : ( ( 1st `  F
) `  z ) --> ( ( 1st `  F
) `  w )
) )
122120, 121mpbid 210 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( z ( 2nd `  F ) w ) `
 h ) : ( ( 1st `  F
) `  z ) --> ( ( 1st `  F
) `  w )
)
123122adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( z ( 2nd `  F
) w ) `  h ) : ( ( 1st `  F
) `  z ) --> ( ( 1st `  F
) `  w )
)
1245, 106, 91, 107, 109, 112, 117, 123setcco 15264 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( ( z ( 2nd `  F
) w ) `  h ) ( <.
( ( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  z
) >. (comp `  S
) ( ( 1st `  F ) `  w
) ) ( ( X ( 2nd `  F
) z ) `  k ) )  =  ( ( ( z ( 2nd `  F
) w ) `  h )  o.  (
( X ( 2nd `  F ) z ) `
 k ) ) )
125101, 104, 1243eqtr3d 2516 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( X ( 2nd `  F
) w ) `  ( k ( <.
w ,  z >.
(comp `  C ) X ) h ) )  =  ( ( ( z ( 2nd `  F ) w ) `
 h )  o.  ( ( X ( 2nd `  F ) z ) `  k
) ) )
126125fveq1d 5866 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( ( X ( 2nd `  F
) w ) `  ( k ( <.
w ,  z >.
(comp `  C ) X ) h ) ) `  A )  =  ( ( ( ( z ( 2nd `  F ) w ) `
 h )  o.  ( ( X ( 2nd `  F ) z ) `  k
) ) `  A
) )
12719ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  A  e.  ( ( 1st `  F
) `  X )
)
128 fvco3 5942 . . . . . . . . . 10  |-  ( ( ( ( X ( 2nd `  F ) z ) `  k
) : ( ( 1st `  F ) `
 X ) --> ( ( 1st `  F
) `  z )  /\  A  e.  (
( 1st `  F
) `  X )
)  ->  ( (
( ( z ( 2nd `  F ) w ) `  h
)  o.  ( ( X ( 2nd `  F
) z ) `  k ) ) `  A )  =  ( ( ( z ( 2nd `  F ) w ) `  h
) `  ( (
( X ( 2nd `  F ) z ) `
 k ) `  A ) ) )
129117, 127, 128syl2anc 661 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( ( ( z ( 2nd `  F ) w ) `
 h )  o.  ( ( X ( 2nd `  F ) z ) `  k
) ) `  A
)  =  ( ( ( z ( 2nd `  F ) w ) `
 h ) `  ( ( ( X ( 2nd `  F
) z ) `  k ) `  A
) ) )
130126, 129eqtrd 2508 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( ( X ( 2nd `  F
) w ) `  ( k ( <.
w ,  z >.
(comp `  C ) X ) h ) ) `  A )  =  ( ( ( z ( 2nd `  F
) w ) `  h ) `  (
( ( X ( 2nd `  F ) z ) `  k
) `  A )
) )
13184adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  C  e.  Cat )
13240, 4oppchom 14967 . . . . . . . . . . . 12  |-  ( z ( Hom  `  O
) w )  =  ( w ( Hom  `  C ) z )
133100, 132syl6eleq 2565 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  h  e.  ( w ( Hom  `  C
) z ) )
1341, 2, 131, 94, 40, 95, 102, 97, 133, 98yon12 15388 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( ( z ( 2nd `  (
( 1st `  Y
) `  X )
) w ) `  h ) `  k
)  =  ( k ( <. w ,  z
>. (comp `  C ) X ) h ) )
135134fveq2d 5868 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( ( ( F N X ) `  A ) `
 w ) `  ( ( ( z ( 2nd `  (
( 1st `  Y
) `  X )
) w ) `  h ) `  k
) )  =  ( ( ( ( F N X ) `  A ) `  w
) `  ( k
( <. w ,  z
>. (comp `  C ) X ) h ) ) )
13613ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  V  e.  W
)
13714ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ran  ( Hom f  `  C
)  C_  U )
13815ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ran  ( Hom f  `  Q )  u.  U
)  C_  V )
13916ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  F  e.  ( O  Func  S )
)
1402, 40, 102, 131, 97, 95, 94, 133, 98catcocl 14936 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( k (
<. w ,  z >.
(comp `  C ) X ) h )  e.  ( w ( Hom  `  C ) X ) )
1411, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 131, 136, 137, 138, 139, 94, 18, 127, 97, 140yonedalem4b 15399 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( ( ( F N X ) `  A ) `
 w ) `  ( k ( <.
w ,  z >.
(comp `  C ) X ) h ) )  =  ( ( ( X ( 2nd `  F ) w ) `
 ( k (
<. w ,  z >.
(comp `  C ) X ) h ) ) `  A ) )
142135, 141eqtrd 2508 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( ( ( F N X ) `  A ) `
 w ) `  ( ( ( z ( 2nd `  (
( 1st `  Y
) `  X )
) w ) `  h ) `  k
) )  =  ( ( ( X ( 2nd `  F ) w ) `  (
k ( <. w ,  z >. (comp `  C ) X ) h ) ) `  A ) )
1431, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 131, 136, 137, 138, 139, 94, 18, 127, 95, 98yonedalem4b 15399 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( ( ( F N X ) `  A ) `
 z ) `  k )  =  ( ( ( X ( 2nd `  F ) z ) `  k
) `  A )
)
144143fveq2d 5868 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( ( z ( 2nd `  F
) w ) `  h ) `  (
( ( ( F N X ) `  A ) `  z
) `  k )
)  =  ( ( ( z ( 2nd `  F ) w ) `
 h ) `  ( ( ( X ( 2nd `  F
) z ) `  k ) `  A
) ) )
145130, 142, 1443eqtr4d 2518 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  ( ( ( ( F N X ) `  A ) `
 w ) `  ( ( ( z ( 2nd `  (
( 1st `  Y
) `  X )
) w ) `  h ) `  k
) )  =  ( ( ( z ( 2nd `  F ) w ) `  h
) `  ( (
( ( F N X ) `  A
) `  z ) `  k ) ) )
14689, 145syldan 470 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  z )
)  ->  ( (
( ( F N X ) `  A
) `  w ) `  ( ( ( z ( 2nd `  (
( 1st `  Y
) `  X )
) w ) `  h ) `  k
) )  =  ( ( ( z ( 2nd `  F ) w ) `  h
) `  ( (
( ( F N X ) `  A
) `  z ) `  k ) ) )
147146mpteq2dva 4533 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
k  e.  ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 z )  |->  ( ( ( ( F N X ) `  A ) `  w
) `  ( (
( z ( 2nd `  ( ( 1st `  Y
) `  X )
) w ) `  h ) `  k
) ) )  =  ( k  e.  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  z )  |->  ( ( ( z ( 2nd `  F
) w ) `  h ) `  (
( ( ( F N X ) `  A ) `  z
) `  k )
) ) )
14827fveq1d 5866 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( F N X ) `  A ) `  z
)  =  ( ( z  e.  B  |->  ( g  e.  ( z ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) z ) `  g
) `  A )
) ) `  z
) )
149 ovex 6307 . . . . . . . . . . . . . 14  |-  ( z ( Hom  `  C
) X )  e. 
_V
150149mptex 6129 . . . . . . . . . . . . 13  |-  ( g  e.  ( z ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) z ) `
 g ) `  A ) )  e. 
_V
151 eqid 2467 . . . . . . . . . . . . . 14  |-  ( z  e.  B  |->  ( g  e.  ( z ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) z ) `
 g ) `  A ) ) )  =  ( z  e.  B  |->  ( g  e.  ( z ( Hom  `  C ) X ) 
|->  ( ( ( X ( 2nd `  F
) z ) `  g ) `  A
) ) )
152151fvmpt2 5955 . . . . . . . . . . . . 13  |-  ( ( z  e.  B  /\  ( g  e.  ( z ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) z ) `  g
) `  A )
)  e.  _V )  ->  ( ( z  e.  B  |->  ( g  e.  ( z ( Hom  `  C ) X ) 
|->  ( ( ( X ( 2nd `  F
) z ) `  g ) `  A
) ) ) `  z )  =  ( g  e.  ( z ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) z ) `  g
) `  A )
) )
153150, 152mpan2 671 . . . . . . . . . . . 12  |-  ( z  e.  B  ->  (
( z  e.  B  |->  ( g  e.  ( z ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) z ) `  g
) `  A )
) ) `  z
)  =  ( g  e.  ( z ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) z ) `
 g ) `  A ) ) )
154148, 153sylan9eq 2528 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  B )  ->  (
( ( F N X ) `  A
) `  z )  =  ( g  e.  ( z ( Hom  `  C ) X ) 
|->  ( ( ( X ( 2nd `  F
) z ) `  g ) `  A
) ) )
155154feq1d 5715 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  B )  ->  (
( ( ( F N X ) `  A ) `  z
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 z ) --> ( ( 1st `  F
) `  z )  <->  ( g  e.  ( z ( Hom  `  C
) X )  |->  ( ( ( X ( 2nd `  F ) z ) `  g
) `  A )
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 z ) --> ( ( 1st `  F
) `  z )
) )
15667, 155mpbird 232 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  (
( ( F N X ) `  A
) `  z ) : ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  z ) --> ( ( 1st `  F
) `  z )
)
157156ralrimiva 2878 . . . . . . . 8  |-  ( ph  ->  A. z  e.  B  ( ( ( F N X ) `  A ) `  z
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 z ) --> ( ( 1st `  F
) `  z )
)
158157adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  A. z  e.  B  ( (
( F N X ) `  A ) `
 z ) : ( ( 1st `  (
( 1st `  Y
) `  X )
) `  z ) --> ( ( 1st `  F
) `  z )
)
159 fveq2 5864 . . . . . . . . 9  |-  ( z  =  w  ->  (
( ( F N X ) `  A
) `  z )  =  ( ( ( F N X ) `
 A ) `  w ) )
160 fveq2 5864 . . . . . . . . 9  |-  ( z  =  w  ->  (
( 1st `  (
( 1st `  Y
) `  X )
) `  z )  =  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  w )
)
161 fveq2 5864 . . . . . . . . 9  |-  ( z  =  w  ->  (
( 1st `  F
) `  z )  =  ( ( 1st `  F ) `  w
) )
162159, 160, 161feq123d 5719 . . . . . . . 8  |-  ( z  =  w  ->  (
( ( ( F N X ) `  A ) `  z
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 z ) --> ( ( 1st `  F
) `  z )  <->  ( ( ( F N X ) `  A
) `  w ) : ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  w ) --> ( ( 1st `  F
) `  w )
) )
163162rspcv 3210 . . . . . . 7  |-  ( w  e.  B  ->  ( A. z  e.  B  ( ( ( F N X ) `  A ) `  z
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 z ) --> ( ( 1st `  F
) `  z )  ->  ( ( ( F N X ) `  A ) `  w
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 w ) --> ( ( 1st `  F
) `  w )
) )
16496, 158, 163sylc 60 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( ( F N X ) `  A
) `  w ) : ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  w ) --> ( ( 1st `  F
) `  w )
)
16570adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  ( 1st `  ( ( 1st `  Y ) `  X
) ) ( O 
Func  S ) ( 2nd `  ( ( 1st `  Y
) `  X )
) )
16628, 29, 30, 165, 86, 96funcf2 15091 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
z ( 2nd `  (
( 1st `  Y
) `  X )
) w ) : ( z ( Hom  `  O ) w ) --> ( ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  z )
( Hom  `  S ) ( ( 1st `  (
( 1st `  Y
) `  X )
) `  w )
) )
167166, 119ffvelrnd 6020 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( z ( 2nd `  ( ( 1st `  Y
) `  X )
) w ) `  h )  e.  ( ( ( 1st `  (
( 1st `  Y
) `  X )
) `  z )
( Hom  `  S ) ( ( 1st `  (
( 1st `  Y
) `  X )
) `  w )
) )
168743ad2antr1 1161 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( 1st `  (
( 1st `  Y
) `  X )
) `  z )  e.  U )
16973adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  ( 1st `  ( ( 1st `  Y ) `  X
) ) : B --> U )
170169, 96ffvelrnd 6020 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( 1st `  (
( 1st `  Y
) `  X )
) `  w )  e.  U )
1715, 105, 30, 168, 170elsetchom 15262 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( ( z ( 2nd `  ( ( 1st `  Y ) `
 X ) ) w ) `  h
)  e.  ( ( ( 1st `  (
( 1st `  Y
) `  X )
) `  z )
( Hom  `  S ) ( ( 1st `  (
( 1st `  Y
) `  X )
) `  w )
)  <->  ( ( z ( 2nd `  (
( 1st `  Y
) `  X )
) w ) `  h ) : ( ( 1st `  (
( 1st `  Y
) `  X )
) `  z ) --> ( ( 1st `  (
( 1st `  Y
) `  X )
) `  w )
) )
172167, 171mpbid 210 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( z ( 2nd `  ( ( 1st `  Y
) `  X )
) w ) `  h ) : ( ( 1st `  (
( 1st `  Y
) `  X )
) `  z ) --> ( ( 1st `  (
( 1st `  Y
) `  X )
) `  w )
)
173 fcompt 6055 . . . . . 6  |-  ( ( ( ( ( F N X ) `  A ) `  w
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 w ) --> ( ( 1st `  F
) `  w )  /\  ( ( z ( 2nd `  ( ( 1st `  Y ) `
 X ) ) w ) `  h
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 z ) --> ( ( 1st `  (
( 1st `  Y
) `  X )
) `  w )
)  ->  ( (
( ( F N X ) `  A
) `  w )  o.  ( ( z ( 2nd `  ( ( 1st `  Y ) `
 X ) ) w ) `  h
) )  =  ( k  e.  ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 z )  |->  ( ( ( ( F N X ) `  A ) `  w
) `  ( (
( z ( 2nd `  ( ( 1st `  Y
) `  X )
) w ) `  h ) `  k
) ) ) )
174164, 172, 173syl2anc 661 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( ( ( F N X ) `  A ) `  w
)  o.  ( ( z ( 2nd `  (
( 1st `  Y
) `  X )
) w ) `  h ) )  =  ( k  e.  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  z )  |->  ( ( ( ( F N X ) `
 A ) `  w ) `  (
( ( z ( 2nd `  ( ( 1st `  Y ) `
 X ) ) w ) `  h
) `  k )
) ) )
1751563ad2antr1 1161 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( ( F N X ) `  A
) `  z ) : ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  z ) --> ( ( 1st `  F
) `  z )
)
176 fcompt 6055 . . . . . 6  |-  ( ( ( ( z ( 2nd `  F ) w ) `  h
) : ( ( 1st `  F ) `
 z ) --> ( ( 1st `  F
) `  w )  /\  ( ( ( F N X ) `  A ) `  z
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 z ) --> ( ( 1st `  F
) `  z )
)  ->  ( (
( z ( 2nd `  F ) w ) `
 h )  o.  ( ( ( F N X ) `  A ) `  z
) )  =  ( k  e.  ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 z )  |->  ( ( ( z ( 2nd `  F ) w ) `  h
) `  ( (
( ( F N X ) `  A
) `  z ) `  k ) ) ) )
177122, 175, 176syl2anc 661 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( ( z ( 2nd `  F ) w ) `  h
)  o.  ( ( ( F N X ) `  A ) `
 z ) )  =  ( k  e.  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  z )  |->  ( ( ( z ( 2nd `  F
) w ) `  h ) `  (
( ( ( F N X ) `  A ) `  z
) `  k )
) ) )
178147, 174, 1773eqtr4d 2518 . . . 4  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( ( ( F N X ) `  A ) `  w
)  o.  ( ( z ( 2nd `  (
( 1st `  Y
) `  X )
) w ) `  h ) )  =  ( ( ( z ( 2nd `  F
) w ) `  h )  o.  (
( ( F N X ) `  A
) `  z )
) )
1795, 105, 91, 168, 170, 111, 172, 164setcco 15264 . . . 4  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( ( ( F N X ) `  A ) `  w
) ( <. (
( 1st `  (
( 1st `  Y
) `  X )
) `  z ) ,  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  w ) >. (comp `  S )
( ( 1st `  F
) `  w )
) ( ( z ( 2nd `  (
( 1st `  Y
) `  X )
) w ) `  h ) )  =  ( ( ( ( F N X ) `
 A ) `  w )  o.  (
( z ( 2nd `  ( ( 1st `  Y
) `  X )
) w ) `  h ) ) )
1805, 105, 91, 168, 108, 111, 175, 122setcco 15264 . . . 4  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( ( z ( 2nd `  F ) w ) `  h
) ( <. (
( 1st `  (
( 1st `  Y
) `  X )
) `  z ) ,  ( ( 1st `  F ) `  z
) >. (comp `  S
) ( ( 1st `  F ) `  w
) ) ( ( ( F N X ) `  A ) `
 z ) )  =  ( ( ( z ( 2nd `  F
) w ) `  h )  o.  (
( ( F N X ) `  A
) `  z )
) )
181178, 179, 1803eqtr4d 2518 . . 3  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( ( ( F N X ) `  A ) `  w
) ( <. (
( 1st `  (
( 1st `  Y
) `  X )
) `  z ) ,  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  w ) >. (comp `  S )
( ( 1st `  F
) `  w )
) ( ( z ( 2nd `  (
( 1st `  Y
) `  X )
) w ) `  h ) )  =  ( ( ( z ( 2nd `  F
) w ) `  h ) ( <.
( ( 1st `  (
( 1st `  Y
) `  X )
) `  z ) ,  ( ( 1st `  F ) `  z
) >. (comp `  S
) ( ( 1st `  F ) `  w
) ) ( ( ( F N X ) `  A ) `
 z ) ) )
182181ralrimivvva 2886 . 2  |-  ( ph  ->  A. z  e.  B  A. w  e.  B  A. h  e.  (
z ( Hom  `  O
) w ) ( ( ( ( F N X ) `  A ) `  w
) ( <. (
( 1st `  (
( 1st `  Y
) `  X )
) `  z ) ,  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  w ) >. (comp `  S )
( ( 1st `  F
) `  w )
) ( ( z ( 2nd `  (
( 1st `  Y
) `  X )
) w ) `  h ) )  =  ( ( ( z ( 2nd `  F
) w ) `  h ) ( <.
( ( 1st `  (
( 1st `  Y
) `  X )
) `  z ) ,  ( ( 1st `  F ) `  z
) >. (comp `  S
) ( ( 1st `  F ) `  w
) ) ( ( ( F N X ) `  A ) `
 z ) ) )
183 eqid 2467 . . 3  |-  ( O Nat 
S )  =  ( O Nat  S )
184183, 28, 29, 30, 91, 68, 16isnat2 15171 . 2  |-  ( ph  ->  ( ( ( F N X ) `  A )  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  <->  ( (
( F N X ) `  A )  e.  X_ z  e.  B  ( ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  z )
( Hom  `  S ) ( ( 1st `  F
) `  z )
)  /\  A. z  e.  B  A. w  e.  B  A. h  e.  ( z ( Hom  `  O ) w ) ( ( ( ( F N X ) `
 A ) `  w ) ( <.
( ( 1st `  (
( 1st `  Y
) `  X )
) `  z ) ,  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  w ) >. (comp `  S )
( ( 1st `  F
) `  w )
) ( ( z ( 2nd `  (
( 1st `  Y
) `  X )
) w ) `  h ) )  =  ( ( ( z ( 2nd `  F
) w ) `  h ) ( <.
( ( 1st `  (
( 1st `  Y
) `  X )
) `  z ) ,  ( ( 1st `  F ) `  z
) >. (comp `  S
) ( ( 1st `  F ) `  w
) ) ( ( ( F N X ) `  A ) `
 z ) ) ) ) )
18583, 182, 184mpbir2and 920 1  |-  ( ph  ->  ( ( F N X ) `  A
)  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    u. cun 3474    C_ wss 3476   <.cop 4033   class class class wbr 4447    |-> cmpt 4505   ran crn 5000    o. ccom 5003   Rel wrel 5004   -->wf 5582   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1stc1st 6779   2ndc2nd 6780  tpos ctpos 6951   X_cixp 7466   Basecbs 14486   Hom chom 14562  compcco 14563   Catccat 14915   Idccid 14916   Hom f chomf 14917  oppCatcoppc 14963    Func cfunc 15077    o.func ccofu 15079   Nat cnat 15164   FuncCat cfuc 15165   SetCatcsetc 15256    X.c cxpc 15291    1stF c1stf 15292    2ndF c2ndf 15293   ⟨,⟩F cprf 15294   evalF cevlf 15332  HomFchof 15371  Yoncyon 15372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-fz 11669  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-hom 14575  df-cco 14576  df-cat 14919  df-cid 14920  df-homf 14921  df-comf 14922  df-oppc 14964  df-func 15081  df-nat 15166  df-fuc 15167  df-setc 15257  df-xpc 15295  df-curf 15337  df-hof 15373  df-yon 15374
This theorem is referenced by:  yonedainv  15404
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