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Theorem yonedalem4c 14329
Description: Lemma for yoneda 14335. (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 )
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  (  Homf  `  C ) 
C_  U )
15 yoneda.v . . . . 5  |-  ( ph  ->  ( ran  (  Homf  `  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 14327 . . . 4  |-  ( ph  ->  ( ( F N X ) `  A
)  =  ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) y ) `
 g ) `  A ) ) ) )
21 oveq1 6047 . . . . . 6  |-  ( y  =  z  ->  (
y (  Hom  `  C
) X )  =  ( z (  Hom  `  C ) X ) )
22 oveq2 6048 . . . . . . . 8  |-  ( y  =  z  ->  ( X ( 2nd `  F
) y )  =  ( X ( 2nd `  F ) z ) )
2322fveq1d 5689 . . . . . . 7  |-  ( y  =  z  ->  (
( X ( 2nd `  F ) y ) `
 g )  =  ( ( X ( 2nd `  F ) z ) `  g
) )
2423fveq1d 5689 . . . . . 6  |-  ( y  =  z  ->  (
( ( X ( 2nd `  F ) y ) `  g
) `  A )  =  ( ( ( X ( 2nd `  F
) z ) `  g ) `  A
) )
2521, 24mpteq12dv 4247 . . . . 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 4260 . . . 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 2452 . . 3  |-  ( ph  ->  ( ( F N X ) `  A
)  =  ( z  e.  B  |->  ( g  e.  ( z (  Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) z ) `
 g ) `  A ) ) ) )
284, 2oppcbas 13899 . . . . . . . . . . . . 13  |-  B  =  ( Base `  O
)
29 eqid 2404 . . . . . . . . . . . . 13  |-  (  Hom  `  O )  =  (  Hom  `  O )
30 eqid 2404 . . . . . . . . . . . . 13  |-  (  Hom  `  S )  =  (  Hom  `  S )
31 relfunc 14014 . . . . . . . . . . . . . . 15  |-  Rel  ( O  Func  S )
32 1st2ndbr 6355 . . . . . . . . . . . . . . 15  |-  ( ( Rel  ( O  Func  S )  /\  F  e.  ( O  Func  S
) )  ->  ( 1st `  F ) ( O  Func  S )
( 2nd `  F
) )
3331, 16, 32sylancr 645 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1st `  F
) ( O  Func  S ) ( 2nd `  F
) )
3433adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  B )  ->  ( 1st `  F ) ( O  Func  S )
( 2nd `  F
) )
3517adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  B )  ->  X  e.  B )
36 simpr 448 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  B )  ->  z  e.  B )
3728, 29, 30, 34, 35, 36funcf2 14020 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  B )  ->  ( X ( 2nd `  F
) z ) : ( X (  Hom  `  O ) z ) --> ( ( ( 1st `  F ) `  X
) (  Hom  `  S
) ( ( 1st `  F ) `  z
) ) )
3837adantr 452 . . . . . . . . . . 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 448 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z (  Hom  `  C ) X ) )  ->  g  e.  ( z (  Hom  `  C ) X ) )
40 eqid 2404 . . . . . . . . . . . . 13  |-  (  Hom  `  C )  =  (  Hom  `  C )
4140, 4oppchom 13896 . . . . . . . . . . . 12  |-  ( X (  Hom  `  O
) z )  =  ( z (  Hom  `  C ) X )
4239, 41syl6eleqr 2495 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z (  Hom  `  C ) X ) )  ->  g  e.  ( X (  Hom  `  O
) z ) )
4338, 42ffvelrnd 5830 . . . . . . . . . 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 3485 . . . . . . . . . . . . . 14  |-  ( ph  ->  U  C_  V )
4513, 44ssexd 4310 . . . . . . . . . . . . 13  |-  ( ph  ->  U  e.  _V )
4645adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  B )  ->  U  e.  _V )
4746adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z (  Hom  `  C ) X ) )  ->  U  e.  _V )
48 eqid 2404 . . . . . . . . . . . . . . 15  |-  ( Base `  S )  =  (
Base `  S )
4928, 48, 33funcf1 14018 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  S ) )
50 eqidd 2405 . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  =  B )
515, 45setcbas 14188 . . . . . . . . . . . . . . 15  |-  ( ph  ->  U  =  ( Base `  S ) )
5250, 51feq23d 5547 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 1st `  F
) : B --> U  <->  ( 1st `  F ) : B --> ( Base `  S )
) )
5349, 52mpbird 224 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1st `  F
) : B --> U )
5453, 17ffvelrnd 5830 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1st `  F
) `  X )  e.  U )
5554ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z (  Hom  `  C ) X ) )  ->  ( ( 1st `  F ) `  X )  e.  U
)
5653ffvelrnda 5829 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  F
) `  z )  e.  U )
5756adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z (  Hom  `  C ) X ) )  ->  ( ( 1st `  F ) `  z )  e.  U
)
585, 47, 30, 55, 57elsetchom 14191 . . . . . . . . . 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 202 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z (  Hom  `  C ) X ) )  ->  ( ( X ( 2nd `  F
) z ) `  g ) : ( ( 1st `  F
) `  X ) --> ( ( 1st `  F
) `  z )
)
6019ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z (  Hom  `  C ) X ) )  ->  A  e.  ( ( 1st `  F
) `  X )
)
6159, 60ffvelrnd 5830 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  B )  /\  g  e.  ( z (  Hom  `  C ) X ) )  ->  ( (
( X ( 2nd `  F ) z ) `
 g ) `  A )  e.  ( ( 1st `  F
) `  z )
)
62 eqid 2404 . . . . . . . 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 5852 . . . . . . 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 452 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  C  e.  Cat )
651, 2, 64, 35, 40, 36yon11 14316 . . . . . . . 8  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  Y
) `  X )
) `  z )  =  ( z (  Hom  `  C ) X ) )
6665feq2d 5540 . . . . . . 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 224 . . . . . 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 14315 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 1st `  Y
) `  X )  e.  ( O  Func  S
) )
69 1st2ndbr 6355 . . . . . . . . . . 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 645 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
) ( O  Func  S ) ( 2nd `  (
( 1st `  Y
) `  X )
) )
7128, 48, 70funcf1 14018 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
) : B --> ( Base `  S ) )
7250, 51feq23d 5547 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) : B --> U  <->  ( 1st `  ( ( 1st `  Y
) `  X )
) : B --> ( Base `  S ) ) )
7371, 72mpbird 224 . . . . . . . 8  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
) : B --> U )
7473ffvelrnda 5829 . . . . . . 7  |-  ( (
ph  /\  z  e.  B )  ->  (
( 1st `  (
( 1st `  Y
) `  X )
) `  z )  e.  U )
755, 46, 30, 74, 56elsetchom 14191 . . . . . 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 224 . . . . 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 2749 . . . 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 5701 . . . . . 6  |-  ( Base `  C )  e.  _V
792, 78eqeltri 2474 . . . . 5  |-  B  e. 
_V
80 mptelixpg 7058 . . . . 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 8 . . . 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 204 . . 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 2478 . 2  |-  ( ph  ->  ( ( F N X ) `  A
)  e.  X_ z  e.  B  ( (
( 1st `  (
( 1st `  Y
) `  X )
) `  z )
(  Hom  `  S ) ( ( 1st `  F
) `  z )
) )
8412adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z (  Hom  `  O ) w ) ) )  ->  C  e.  Cat )
8517adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z (  Hom  `  O ) w ) ) )  ->  X  e.  B )
86 simpr1 963 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z (  Hom  `  O ) w ) ) )  ->  z  e.  B )
871, 2, 84, 85, 40, 86yon11 14316 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z (  Hom  `  O ) w ) ) )  ->  (
( 1st `  (
( 1st `  Y
) `  X )
) `  z )  =  ( z (  Hom  `  C ) X ) )
8887eleq2d 2471 . . . . . . . 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 471 . . . . . . 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 2404 . . . . . . . . . . . 12  |-  (comp `  O )  =  (comp `  O )
91 eqid 2404 . . . . . . . . . . . 12  |-  (comp `  S )  =  (comp `  S )
9233adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z (  Hom  `  O ) w ) ) )  ->  ( 1st `  F ) ( O  Func  S )
( 2nd `  F
) )
9392adantr 452 . . . . . . . . . . . 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 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
(  Hom  `  O ) w ) ) )  /\  k  e.  ( z (  Hom  `  C
) X ) )  ->  X  e.  B
)
9586adantr 452 . . . . . . . . . . . 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 964 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z (  Hom  `  O ) w ) ) )  ->  w  e.  B )
9796adantr 452 . . . . . . . . . . . 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 448 . . . . . . . . . . . . 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 2495 . . . . . . . . . . . 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 1001 . . . . . . . . . . . 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 14023 . . . . . . . . . . 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 2404 . . . . . . . . . . . . 13  |-  (comp `  C )  =  (comp `  C )
1032, 102, 4, 94, 95, 97oppcco 13898 . . . . . . . . . . . 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 5691 . . . . . . . . . . 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 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z (  Hom  `  O ) w ) ) )  ->  U  e.  _V )
106105adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
(  Hom  `  O ) w ) ) )  /\  k  e.  ( z (  Hom  `  C
) X ) )  ->  U  e.  _V )
10754ad2antrr 707 . . . . . . . . . . . 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 1122 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z (  Hom  `  O ) w ) ) )  ->  (
( 1st `  F
) `  z )  e.  U )
109108adantr 452 . . . . . . . . . . . 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 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z (  Hom  `  O ) w ) ) )  ->  ( 1st `  F ) : B --> U )
111110, 96ffvelrnd 5830 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z (  Hom  `  O ) w ) ) )  ->  (
( 1st `  F
) `  w )  e.  U )
112111adantr 452 . . . . . . . . . . . 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 14020 . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . 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 5830 . . . . . . . . . . . . 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 14191 . . . . . . . . . . . . 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 202 . . . . . . . . . . . 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 14020 . . . . . . . . . . . . . . 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 965 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z (  Hom  `  O ) w ) ) )  ->  h  e.  ( z (  Hom  `  O ) w ) )
120118, 119ffvelrnd 5830 . . . . . . . . . . . . . 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 14191 . . . . . . . . . . . . . 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 202 . . . . . . . . . . . . 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 452 . . . . . . . . . . . 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 14193 . . . . . . . . . . 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 2444 . . . . . . . . . 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 5689 . . . . . . . . 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 707 . . . . . . . . . 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 5759 . . . . . . . . . 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 643 . . . . . . . . 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 2436 . . . . . . . 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 452 . . . . . . . . . . 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 13896 . . . . . . . . . . . 12  |-  ( z (  Hom  `  O
) w )  =  ( w (  Hom  `  C ) z )
133100, 132syl6eleq 2494 . . . . . . . . . . 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 14317 . . . . . . . . . 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 5691 . . . . . . . . 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 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
(  Hom  `  O ) w ) ) )  /\  k  e.  ( z (  Hom  `  C
) X ) )  ->  V  e.  W
)
13714ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
(  Hom  `  O ) w ) ) )  /\  k  e.  ( z (  Hom  `  C
) X ) )  ->  ran  (  Homf  `  C
)  C_  U )
13815ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B  /\  h  e.  ( z
(  Hom  `  O ) w ) ) )  /\  k  e.  ( z (  Hom  `  C
) X ) )  ->  ( ran  (  Homf  `  Q )  u.  U
)  C_  V )
13916ad2antrr 707 . . . . . . . . . 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 13865 . . . . . . . . . 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 14328 . . . . . . . . 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 2436 . . . . . . . 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 14328 . . . . . . . . 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 5691 . . . . . . . 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 2446 . . . . . . 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 457 . . . . . 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 4255 . . . . 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 5689 . . . . . . . . . . . 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 6065 . . . . . . . . . . . . . 14  |-  ( z (  Hom  `  C
) X )  e. 
_V
150149mptex 5925 . . . . . . . . . . . . 13  |-  ( g  e.  ( z (  Hom  `  C ) X )  |->  ( ( ( X ( 2nd `  F ) z ) `
 g ) `  A ) )  e. 
_V
151 eqid 2404 . . . . . . . . . . . . . 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 5771 . . . . . . . . . . . . 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 653 . . . . . . . . . . . 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 2456 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  B )  ->  (
( ( F N X ) `  A
) `  z )  =  ( g  e.  ( z (  Hom  `  C ) X ) 
|->  ( ( ( X ( 2nd `  F
) z ) `  g ) `  A
) ) )
155154feq1d 5539 . . . . . . . . . 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 224 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  B )  ->  (
( ( F N X ) `  A
) `  z ) : ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  z ) --> ( ( 1st `  F
) `  z )
)
157156ralrimiva 2749 . . . . . . . 8  |-  ( ph  ->  A. z  e.  B  ( ( ( F N X ) `  A ) `  z
) : ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 z ) --> ( ( 1st `  F
) `  z )
)
158157adantr 452 . . . . . . 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 5687 . . . . . . . . 9  |-  ( z  =  w  ->  (
( ( F N X ) `  A
) `  z )  =  ( ( ( F N X ) `
 A ) `  w ) )
160 fveq2 5687 . . . . . . . . 9  |-  ( z  =  w  ->  (
( 1st `  (
( 1st `  Y
) `  X )
) `  z )  =  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  w )
)
161 fveq2 5687 . . . . . . . . 9  |-  ( z  =  w  ->  (
( 1st `  F
) `  z )  =  ( ( 1st `  F ) `  w
) )
162159, 160, 161feq123d 5542 . . . . . . . 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 3008 . . . . . . 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 58 . . . . . 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 452 . . . . . . . . 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 14020 . . . . . . . 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 5830 . . . . . . 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 1122 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z (  Hom  `  O ) w ) ) )  ->  (
( 1st `  (
( 1st `  Y
) `  X )
) `  z )  e.  U )
16973adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B  /\  h  e.  ( z (  Hom  `  O ) w ) ) )  ->  ( 1st `  ( ( 1st `  Y ) `  X
) ) : B --> U )
170169, 96ffvelrnd 5830 . . . . . . . 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 14191 . . . . . . 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 202 . . . . . 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 5863 . . . . . 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 643 . . . . 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 1122 . . . . . 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 5863 . . . . . 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 643 . . . . 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 2446 . . . 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 14193 . . . 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 14193 . . . 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 2446 . . 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 2759 . 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 2404 . . 3  |-  ( O Nat 
S )  =  ( O Nat  S )
184183, 28, 29, 30, 91, 68, 16isnat2 14100 . 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 889 1  |-  ( ph  ->  ( ( F N X ) `  A
)  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    u. cun 3278    C_ wss 3280   <.cop 3777   class class class wbr 4172    e. cmpt 4226   ran crn 4838    o. ccom 4841   Rel wrel 4842   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307  tpos ctpos 6437   X_cixp 7022   Basecbs 13424    Hom chom 13495  compcco 13496   Catccat 13844   Idccid 13845    Homf chomf 13846  oppCatcoppc 13892    Func cfunc 14006    o.func ccofu 14008   Nat cnat 14093   FuncCat cfuc 14094   SetCatcsetc 14185    X.c cxpc 14220    1stF c1stf 14221    2ndF c2ndf 14222   ⟨,⟩F cprf 14223   evalF cevlf 14261  HomFchof 14300  Yoncyon 14301
This theorem is referenced by:  yonedainv  14333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-hom 13508  df-cco 13509  df-cat 13848  df-cid 13849  df-homf 13850  df-comf 13851  df-oppc 13893  df-func 14010  df-nat 14095  df-fuc 14096  df-setc 14186  df-xpc 14224  df-curf 14266  df-hof 14302  df-yon 14303
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