MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  yonedalem4c Structured version   Unicode version

Theorem yonedalem4c 15083
Description: Lemma for yoneda 15089. (Contributed by Mario Carneiro, 29-Jan-2017.)
Hypotheses
Ref Expression
yoneda.y  |-  Y  =  (Yon `  C )
yoneda.b  |-  B  =  ( Base `  C
)
yoneda.1  |-  .1.  =  ( Id `  C )
yoneda.o  |-  O  =  (oppCat `  C )
yoneda.s  |-  S  =  ( SetCat `  U )
yoneda.t  |-  T  =  ( SetCat `  V )
yoneda.q  |-  Q  =  ( O FuncCat  S )
yoneda.h  |-  H  =  (HomF
`  Q )
yoneda.r  |-  R  =  ( ( Q  X.c  O
) FuncCat  T )
yoneda.e  |-  E  =  ( O evalF  S )
yoneda.z  |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )
yoneda.c  |-  ( ph  ->  C  e.  Cat )
yoneda.w  |-  ( ph  ->  V  e.  W )
yoneda.u  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
yoneda.v  |-  ( ph  ->  ( ran  ( Hom f  `  Q )  u.  U
)  C_  V )
yonedalem21.f  |-  ( ph  ->  F  e.  ( O 
Func  S ) )
yonedalem21.x  |-  ( ph  ->  X  e.  B )
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 15081 . . . 4  |-  ( ph  ->  ( ( F N X ) `  A
)  =  ( y  e.  B  |->  ( g  e.  ( y ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A ) ) ) )
21 oveq1 6097 . . . . . 6  |-  ( y  =  z  ->  (
y ( Hom  `  C
) X )  =  ( z ( Hom  `  C ) X ) )
22 oveq2 6098 . . . . . . . 8  |-  ( y  =  z  ->  ( X ( 2nd `  F
) y )  =  ( X ( 2nd `  F ) z ) )
2322fveq1d 5690 . . . . . . 7  |-  ( y  =  z  ->  (
( X ( 2nd `  F ) y ) `
 g )  =  ( ( X ( 2nd `  F ) z ) `  g
) )
2423fveq1d 5690 . . . . . 6  |-  ( y  =  z  ->  (
( ( X ( 2nd `  F ) y ) `  g
) `  A )  =  ( ( ( X ( 2nd `  F
) z ) `  g ) `  A
) )
2521, 24mpteq12dv 4367 . . . . 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 4380 . . . 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 2489 . . 3  |-  ( ph  ->  ( ( F N X ) `  A
)  =  ( z  e.  B  |->  ( g  e.  ( z ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) z ) `
 g ) `  A ) ) ) )
284, 2oppcbas 14653 . . . . . . . . . . . . 13  |-  B  =  ( Base `  O
)
29 eqid 2441 . . . . . . . . . . . . 13  |-  ( Hom  `  O )  =  ( Hom  `  O )
30 eqid 2441 . . . . . . . . . . . . 13  |-  ( Hom  `  S )  =  ( Hom  `  S )
31 relfunc 14768 . . . . . . . . . . . . . . 15  |-  Rel  ( O  Func  S )
32 1st2ndbr 6622 . . . . . . . . . . . . . . 15  |-  ( ( Rel  ( O  Func  S )  /\  F  e.  ( O  Func  S
) )  ->  ( 1st `  F ) ( O  Func  S )
( 2nd `  F
) )
3331, 16, 32sylancr 658 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1st `  F
) ( O  Func  S ) ( 2nd `  F
) )
3433adantr 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  B )  ->  ( 1st `  F ) ( O  Func  S )
( 2nd `  F
) )
3517adantr 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  B )  ->  X  e.  B )
36 simpr 458 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  B )  ->  z  e.  B )
3728, 29, 30, 34, 35, 36funcf2 14774 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  B )  ->  ( X ( 2nd `  F
) z ) : ( X ( Hom  `  O ) z ) --> ( ( ( 1st `  F ) `  X
) ( Hom  `  S
) ( ( 1st `  F ) `  z
) ) )
3837adantr 462 . . . . . . . . . . 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 458 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z ( Hom  `  C ) X ) )  ->  g  e.  ( z ( Hom  `  C ) X ) )
40 eqid 2441 . . . . . . . . . . . . 13  |-  ( Hom  `  C )  =  ( Hom  `  C )
4140, 4oppchom 14650 . . . . . . . . . . . 12  |-  ( X ( Hom  `  O
) z )  =  ( z ( Hom  `  C ) X )
4239, 41syl6eleqr 2532 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z ( Hom  `  C ) X ) )  ->  g  e.  ( X ( Hom  `  O
) z ) )
4338, 42ffvelrnd 5841 . . . . . . . . . 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 3531 . . . . . . . . . . . . . 14  |-  ( ph  ->  U  C_  V )
4513, 44ssexd 4436 . . . . . . . . . . . . 13  |-  ( ph  ->  U  e.  _V )
4645adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  B )  ->  U  e.  _V )
4746adantr 462 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z ( Hom  `  C ) X ) )  ->  U  e.  _V )
48 eqid 2441 . . . . . . . . . . . . . . 15  |-  ( Base `  S )  =  (
Base `  S )
4928, 48, 33funcf1 14772 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  S ) )
50 eqidd 2442 . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  =  B )
515, 45setcbas 14942 . . . . . . . . . . . . . . 15  |-  ( ph  ->  U  =  ( Base `  S ) )
5250, 51feq23d 5551 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 1st `  F
) : B --> U  <->  ( 1st `  F ) : B --> ( Base `  S )
) )
5349, 52mpbird 232 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1st `  F
) : B --> U )
5453, 17ffvelrnd 5841 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1st `  F
) `  X )  e.  U )
5554ad2antrr 720 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z ( Hom  `  C ) X ) )  ->  ( ( 1st `  F ) `  X )  e.  U
)
5653ffvelrnda 5840 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  F
) `  z )  e.  U )
5756adantr 462 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z ( Hom  `  C ) X ) )  ->  ( ( 1st `  F ) `  z )  e.  U
)
585, 47, 30, 55, 57elsetchom 14945 . . . . . . . . . 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 720 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z ( Hom  `  C ) X ) )  ->  A  e.  ( ( 1st `  F
) `  X )
)
6159, 60ffvelrnd 5841 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z ( Hom  `  C ) X ) )  ->  ( (
( X ( 2nd `  F ) z ) `
 g ) `  A )  e.  ( ( 1st `  F
) `  z )
)
62 eqid 2441 . . . . . . . 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 5864 . . . . . . 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 462 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  C  e.  Cat )
651, 2, 64, 35, 40, 36yon11 15070 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  Y
) `  X )
) `  z )  =  ( z ( Hom  `  C ) X ) )
6665feq2d 5544 . . . . . . 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 15069 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 1st `  Y
) `  X )  e.  ( O  Func  S
) )
69 1st2ndbr 6622 . . . . . . . . . . 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 658 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
) ( O  Func  S ) ( 2nd `  (
( 1st `  Y
) `  X )
) )
7128, 48, 70funcf1 14772 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
) : B --> ( Base `  S ) )
7250, 51feq23d 5551 . . . . . . . . 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 5840 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  Y
) `  X )
) `  z )  e.  U )
755, 46, 30, 74, 56elsetchom 14945 . . . . . 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 2797 . . . 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 5698 . . . . . 6  |-  ( Base `  C )  e.  _V
792, 78eqeltri 2511 . . . . 5  |-  B  e. 
_V
80 mptelixpg 7296 . . . . 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 2515 . 2  |-  ( ph  ->  ( ( F N X ) `  A
)  e.  X_ z  e.  B  ( (
( 1st `  (
( 1st `  Y
) `  X )
) `  z )
( Hom  `  S ) ( ( 1st `  F
) `  z )
) )
8412adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  C  e.  Cat )
8517adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  X  e.  B )
86 simpr1 989 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  z  e.  B )
871, 2, 84, 85, 40, 86yon11 15070 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( 1st `  (
( 1st `  Y
) `  X )
) `  z )  =  ( z ( Hom  `  C ) X ) )
8887eleq2d 2508 . . . . . . . 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 481 . . . . . . 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 2441 . . . . . . . . . . . 12  |-  (comp `  O )  =  (comp `  O )
91 eqid 2441 . . . . . . . . . . . 12  |-  (comp `  S )  =  (comp `  S )
9233adantr 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  ( 1st `  F ) ( O  Func  S )
( 2nd `  F
) )
9392adantr 462 . . . . . . . . . . . 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 462 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  X  e.  B
)
9586adantr 462 . . . . . . . . . . . 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 990 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  w  e.  B )
9796adantr 462 . . . . . . . . . . . 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 458 . . . . . . . . . . . . 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 2532 . . . . . . . . . . . 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 1027 . . . . . . . . . . . 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 14777 . . . . . . . . . . 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 2441 . . . . . . . . . . . . 13  |-  (comp `  C )  =  (comp `  C )
1032, 102, 4, 94, 95, 97oppcco 14652 . . . . . . . . . . . 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 5692 . . . . . . . . . . 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 462 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  U  e.  _V )
106105adantr 462 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  U  e.  _V )
10754ad2antrr 720 . . . . . . . . . . . 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 1148 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( 1st `  F
) `  z )  e.  U )
109108adantr 462 . . . . . . . . . . . 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 462 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  ( 1st `  F ) : B --> U )
111110, 96ffvelrnd 5841 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( 1st `  F
) `  w )  e.  U )
112111adantr 462 . . . . . . . . . . . 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 14774 . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . 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 5841 . . . . . . . . . . . . 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 14945 . . . . . . . . . . . . 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 14774 . . . . . . . . . . . . . . 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 991 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  h  e.  ( z ( Hom  `  O ) w ) )
120118, 119ffvelrnd 5841 . . . . . . . . . . . . . 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 14945 . . . . . . . . . . . . . 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 462 . . . . . . . . . . . 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 14947 . . . . . . . . . . 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 2481 . . . . . . . . . 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 5690 . . . . . . . . 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 720 . . . . . . . . . 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 5765 . . . . . . . . . 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 656 . . . . . . . . 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 2473 . . . . . . . 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 462 . . . . . . . . . . 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 14650 . . . . . . . . . . . 12  |-  ( z ( Hom  `  O
) w )  =  ( w ( Hom  `  C ) z )
133100, 132syl6eleq 2531 . . . . . . . . . . 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 15071 . . . . . . . . . 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 5692 . . . . . . . . 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 720 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
( Hom  `  O ) w ) ) )  /\  k  e.  ( z ( Hom  `  C
) X ) )  ->  V  e.  W
)
13714ad2antrr 720 . . . . . . . . . 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 720 . . . . . . . . . 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 720 . . . . . . . . . 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 14619 . . . . . . . . . 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 15082 . . . . . . . . 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 2473 . . . . . . . 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 15082 . . . . . . . . 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 5692 . . . . . . . 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 2483 . . . . . . 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 467 . . . . . 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 4375 . . . . 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 5690 . . . . . . . . . . . 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 6115 . . . . . . . . . . . . . 14  |-  ( z ( Hom  `  C
) X )  e. 
_V
150149mptex 5945 . . . . . . . . . . . . 13  |-  ( g  e.  ( z ( Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) z ) `
 g ) `  A ) )  e. 
_V
151 eqid 2441 . . . . . . . . . . . . . 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 5778 . . . . . . . . . . . . 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 666 . . . . . . . . . . . 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 2493 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  B )  ->  (
( ( F N X ) `  A
) `  z )  =  ( g  e.  ( z ( Hom  `  C ) X ) 
|->  ( ( ( X ( 2nd `  F
) z ) `  g ) `  A
) ) )
155154feq1d 5543 . . . . . . . . . 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 2797 . . . . . . . 8  |-  ( ph  ->  A. z  e.  B  ( ( ( F N X ) `  A ) `  z
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 z ) --> ( ( 1st `  F
) `  z )
)
158157adantr 462 . . . . . . 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 5688 . . . . . . . . 9  |-  ( z  =  w  ->  (
( ( F N X ) `  A
) `  z )  =  ( ( ( F N X ) `
 A ) `  w ) )
160 fveq2 5688 . . . . . . . . 9  |-  ( z  =  w  ->  (
( 1st `  (
( 1st `  Y
) `  X )
) `  z )  =  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  w )
)
161 fveq2 5688 . . . . . . . . 9  |-  ( z  =  w  ->  (
( 1st `  F
) `  z )  =  ( ( 1st `  F ) `  w
) )
162159, 160, 161feq123d 5546 . . . . . . . 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 3066 . . . . . . 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 462 . . . . . . . . 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 14774 . . . . . . . 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 5841 . . . . . . 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 1148 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  (
( 1st `  (
( 1st `  Y
) `  X )
) `  z )  e.  U )
16973adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z ( Hom  `  O ) w ) ) )  ->  ( 1st `  ( ( 1st `  Y ) `  X
) ) : B --> U )
170169, 96ffvelrnd 5841 . . . . . . . 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 14945 . . . . . . 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 5876 . . . . . 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 656 . . . . 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 1148 . . . . . 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 5876 . . . . . 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 656 . . . . 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 2483 . . . 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 14947 . . . 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 14947 . . . 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 2483 . . 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 2807 . 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 2441 . . 3  |-  ( O Nat 
S )  =  ( O Nat  S )
184183, 28, 29, 30, 91, 68, 16isnat2 14854 . 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 908 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 960    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970    u. cun 3323    C_ wss 3325   <.cop 3880   class class class wbr 4289    e. cmpt 4347   ran crn 4837    o. ccom 4840   Rel wrel 4841   -->wf 5411   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   1stc1st 6574   2ndc2nd 6575  tpos ctpos 6743   X_cixp 7259   Basecbs 14170   Hom chom 14245  compcco 14246   Catccat 14598   Idccid 14599   Hom f chomf 14600  oppCatcoppc 14646    Func cfunc 14760    o.func ccofu 14762   Nat cnat 14847   FuncCat cfuc 14848   SetCatcsetc 14939    X.c cxpc 14974    1stF c1stf 14975    2ndF c2ndf 14976   ⟨,⟩F cprf 14977   evalF cevlf 15015  HomFchof 15054  Yoncyon 15055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-hom 14258  df-cco 14259  df-cat 14602  df-cid 14603  df-homf 14604  df-comf 14605  df-oppc 14647  df-func 14764  df-nat 14849  df-fuc 14850  df-setc 14940  df-xpc 14978  df-curf 15020  df-hof 15056  df-yon 15057
This theorem is referenced by:  yonedainv  15087
  Copyright terms: Public domain W3C validator