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Theorem yonffthlem 14384
Description: Lemma for yonffth 14386. (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 )
yoneda.m  |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y
) `  x )
( O Nat  S ) f )  |->  ( ( a `  x ) `
 (  .1.  `  x ) ) ) )
yonedainv.i  |-  I  =  (Inv `  R )
yonedainv.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 )
) ) ) )
Assertion
Ref Expression
yonffthlem  |-  ( ph  ->  Y  e.  ( ( C Full  Q )  i^i  ( C Faith  Q ) ) )
Distinct variable groups:    f, a,
g, x, y,  .1.    u, a, g, y, C, f, x    E, a, f, g, u, y    B, a, f, g, u, x, y    N, a    O, a, f, g, u, x, y    S, a, f, g, u, x, y    g, M, u, y    Q, a, f, g, u, x    T, f, g, u, y    ph, a,
f, g, u, x, y    u, R    Y, a, f, g, u, x, y    Z, a, f, g, u, x, y
Allowed substitution hints:    Q( y)    R( x, y, f, g, a)    T( x, a)    U( x, y, u, f, g, a)    .1. ( u)    E( x)    H( x, y, u, f, g, a)    I( x, y, u, f, g, a)    M( x, f, a)    N( x, y, u, f, g)    V( x, y, u, f, g, a)    W( x, y, u, f, g, a)

Proof of Theorem yonffthlem
Dummy variables  h  w  z  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfunc 14064 . . 3  |-  Rel  ( C  Func  Q )
2 yoneda.y . . . 4  |-  Y  =  (Yon `  C )
3 yoneda.c . . . 4  |-  ( ph  ->  C  e.  Cat )
4 yoneda.o . . . 4  |-  O  =  (oppCat `  C )
5 yoneda.s . . . 4  |-  S  =  ( SetCat `  U )
6 yoneda.q . . . 4  |-  Q  =  ( O FuncCat  S )
7 yoneda.w . . . . 5  |-  ( ph  ->  V  e.  W )
8 yoneda.v . . . . . 6  |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U
)  C_  V )
98unssbd 3527 . . . . 5  |-  ( ph  ->  U  C_  V )
107, 9ssexd 4353 . . . 4  |-  ( ph  ->  U  e.  _V )
11 yoneda.u . . . 4  |-  ( ph  ->  ran  (  Homf  `  C ) 
C_  U )
122, 3, 4, 5, 6, 10, 11yoncl 14364 . . 3  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
13 1st2nd 6396 . . 3  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  Y  =  <. ( 1st `  Y
) ,  ( 2nd `  Y ) >. )
141, 12, 13sylancr 646 . 2  |-  ( ph  ->  Y  =  <. ( 1st `  Y ) ,  ( 2nd `  Y
) >. )
15 1st2ndbr 6399 . . . . 5  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  ( 1st `  Y ) ( C  Func  Q )
( 2nd `  Y
) )
161, 12, 15sylancr 646 . . . 4  |-  ( ph  ->  ( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
17 yoneda.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  C
)
186fucbas 14162 . . . . . . . . . . . . 13  |-  ( O 
Func  S )  =  (
Base `  Q )
1917, 18, 16funcf1 14068 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st `  Y
) : B --> ( O 
Func  S ) )
2019adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( 1st `  Y
) : B --> ( O 
Func  S ) )
21 simprr 735 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  w  e.  B )
2220, 21ffvelrnd 5874 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  Y
) `  w )  e.  ( O  Func  S
) )
23 simprl 734 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
z  e.  B )
24 opelxpi 4913 . . . . . . . . . 10  |-  ( ( ( ( 1st `  Y
) `  w )  e.  ( O  Func  S
)  /\  z  e.  B )  ->  <. (
( 1st `  Y
) `  w ) ,  z >.  e.  ( ( O  Func  S
)  X.  B ) )
2522, 23, 24syl2anc 644 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  <. ( ( 1st `  Y
) `  w ) ,  z >.  e.  ( ( O  Func  S
)  X.  B ) )
26 yoneda.r . . . . . . . . . . . . . 14  |-  R  =  ( ( Q  X.c  O
) FuncCat  T )
2726fucbas 14162 . . . . . . . . . . . . 13  |-  ( ( Q  X.c  O )  Func  T
)  =  ( Base `  R )
28 yonedainv.i . . . . . . . . . . . . 13  |-  I  =  (Inv `  R )
29 yoneda.1 . . . . . . . . . . . . . . . . . 18  |-  .1.  =  ( Id `  C )
30 yoneda.t . . . . . . . . . . . . . . . . . 18  |-  T  =  ( SetCat `  V )
31 yoneda.h . . . . . . . . . . . . . . . . . 18  |-  H  =  (HomF
`  Q )
32 yoneda.e . . . . . . . . . . . . . . . . . 18  |-  E  =  ( O evalF  S )
33 yoneda.z . . . . . . . . . . . . . . . . . 18  |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )
342, 17, 29, 4, 5, 30, 6, 31, 26, 32, 33, 3, 7, 11, 8yonedalem1 14374 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( Z  e.  ( ( Q  X.c  O ) 
Func  T )  /\  E  e.  ( ( Q  X.c  O
)  Func  T )
) )
3534simpld 447 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  Z  e.  ( ( Q  X.c  O )  Func  T
) )
36 funcrcl 14065 . . . . . . . . . . . . . . . 16  |-  ( Z  e.  ( ( Q  X.c  O )  Func  T
)  ->  ( ( Q  X.c  O )  e.  Cat  /\  T  e.  Cat )
)
3735, 36syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( Q  X.c  O
)  e.  Cat  /\  T  e.  Cat )
)
3837simpld 447 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( Q  X.c  O )  e.  Cat )
3937simprd 451 . . . . . . . . . . . . . 14  |-  ( ph  ->  T  e.  Cat )
4026, 38, 39fuccat 14172 . . . . . . . . . . . . 13  |-  ( ph  ->  R  e.  Cat )
4134simprd 451 . . . . . . . . . . . . 13  |-  ( ph  ->  E  e.  ( ( Q  X.c  O )  Func  T
) )
42 eqid 2438 . . . . . . . . . . . . 13  |-  (  Iso  `  R )  =  (  Iso  `  R )
43 yoneda.m . . . . . . . . . . . . . 14  |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y
) `  x )
( O Nat  S ) f )  |->  ( ( a `  x ) `
 (  .1.  `  x ) ) ) )
44 yonedainv.n . . . . . . . . . . . . . 14  |-  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 )
) ) ) )
452, 17, 29, 4, 5, 30, 6, 31, 26, 32, 33, 3, 7, 11, 8, 43, 28, 44yonedainv 14383 . . . . . . . . . . . . 13  |-  ( ph  ->  M ( Z I E ) N )
4627, 28, 40, 35, 41, 42, 45inviso2 13997 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  ( E (  Iso  `  R
) Z ) )
47 eqid 2438 . . . . . . . . . . . . . 14  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
484, 17oppcbas 13949 . . . . . . . . . . . . . 14  |-  B  =  ( Base `  O
)
4947, 18, 48xpcbas 14280 . . . . . . . . . . . . 13  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
50 eqid 2438 . . . . . . . . . . . . 13  |-  ( ( Q  X.c  O ) Nat  T )  =  ( ( Q  X.c  O ) Nat  T )
51 eqid 2438 . . . . . . . . . . . . 13  |-  (  Iso  `  T )  =  (  Iso  `  T )
5226, 49, 50, 41, 35, 42, 51fuciso 14177 . . . . . . . . . . . 12  |-  ( ph  ->  ( N  e.  ( E (  Iso  `  R
) Z )  <->  ( N  e.  ( E ( ( Q  X.c  O ) Nat  T ) Z )  /\  A. v  e.  ( ( O  Func  S )  X.  B ) ( N `
 v )  e.  ( ( ( 1st `  E ) `  v
) (  Iso  `  T
) ( ( 1st `  Z ) `  v
) ) ) ) )
5346, 52mpbid 203 . . . . . . . . . . 11  |-  ( ph  ->  ( N  e.  ( E ( ( Q  X.c  O ) Nat  T ) Z )  /\  A. v  e.  ( ( O  Func  S )  X.  B ) ( N `
 v )  e.  ( ( ( 1st `  E ) `  v
) (  Iso  `  T
) ( ( 1st `  Z ) `  v
) ) ) )
5453simprd 451 . . . . . . . . . 10  |-  ( ph  ->  A. v  e.  ( ( O  Func  S
)  X.  B ) ( N `  v
)  e.  ( ( ( 1st `  E
) `  v )
(  Iso  `  T ) ( ( 1st `  Z
) `  v )
) )
5554adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  A. v  e.  (
( O  Func  S
)  X.  B ) ( N `  v
)  e.  ( ( ( 1st `  E
) `  v )
(  Iso  `  T ) ( ( 1st `  Z
) `  v )
) )
56 fveq2 5731 . . . . . . . . . . . 12  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( N `  v
)  =  ( N `
 <. ( ( 1st `  Y ) `  w
) ,  z >.
) )
57 df-ov 6087 . . . . . . . . . . . 12  |-  ( ( ( 1st `  Y
) `  w ) N z )  =  ( N `  <. ( ( 1st `  Y
) `  w ) ,  z >. )
5856, 57syl6eqr 2488 . . . . . . . . . . 11  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( N `  v
)  =  ( ( ( 1st `  Y
) `  w ) N z ) )
59 fveq2 5731 . . . . . . . . . . . . 13  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( ( 1st `  E
) `  v )  =  ( ( 1st `  E ) `  <. ( ( 1st `  Y
) `  w ) ,  z >. )
)
60 df-ov 6087 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  Y
) `  w )
( 1st `  E
) z )  =  ( ( 1st `  E
) `  <. ( ( 1st `  Y ) `
 w ) ,  z >. )
6159, 60syl6eqr 2488 . . . . . . . . . . . 12  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( ( 1st `  E
) `  v )  =  ( ( ( 1st `  Y ) `
 w ) ( 1st `  E ) z ) )
62 fveq2 5731 . . . . . . . . . . . . 13  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( ( 1st `  Z
) `  v )  =  ( ( 1st `  Z ) `  <. ( ( 1st `  Y
) `  w ) ,  z >. )
)
63 df-ov 6087 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  Y
) `  w )
( 1st `  Z
) z )  =  ( ( 1st `  Z
) `  <. ( ( 1st `  Y ) `
 w ) ,  z >. )
6462, 63syl6eqr 2488 . . . . . . . . . . . 12  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( ( 1st `  Z
) `  v )  =  ( ( ( 1st `  Y ) `
 w ) ( 1st `  Z ) z ) )
6561, 64oveq12d 6102 . . . . . . . . . . 11  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( ( ( 1st `  E ) `  v
) (  Iso  `  T
) ( ( 1st `  Z ) `  v
) )  =  ( ( ( ( 1st `  Y ) `  w
) ( 1st `  E
) z ) (  Iso  `  T )
( ( ( 1st `  Y ) `  w
) ( 1st `  Z
) z ) ) )
6658, 65eleq12d 2506 . . . . . . . . . 10  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( ( N `  v )  e.  ( ( ( 1st `  E
) `  v )
(  Iso  `  T ) ( ( 1st `  Z
) `  v )
)  <->  ( ( ( 1st `  Y ) `
 w ) N z )  e.  ( ( ( ( 1st `  Y ) `  w
) ( 1st `  E
) z ) (  Iso  `  T )
( ( ( 1st `  Y ) `  w
) ( 1st `  Z
) z ) ) ) )
6766rspcv 3050 . . . . . . . . 9  |-  ( <.
( ( 1st `  Y
) `  w ) ,  z >.  e.  ( ( O  Func  S
)  X.  B )  ->  ( A. v  e.  ( ( O  Func  S )  X.  B ) ( N `  v
)  e.  ( ( ( 1st `  E
) `  v )
(  Iso  `  T ) ( ( 1st `  Z
) `  v )
)  ->  ( (
( 1st `  Y
) `  w ) N z )  e.  ( ( ( ( 1st `  Y ) `
 w ) ( 1st `  E ) z ) (  Iso  `  T ) ( ( ( 1st `  Y
) `  w )
( 1st `  Z
) z ) ) ) )
6825, 55, 67sylc 59 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  w
) N z )  e.  ( ( ( ( 1st `  Y
) `  w )
( 1st `  E
) z ) (  Iso  `  T )
( ( ( 1st `  Y ) `  w
) ( 1st `  Z
) z ) ) )
694oppccat 13953 . . . . . . . . . . . . 13  |-  ( C  e.  Cat  ->  O  e.  Cat )
703, 69syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  O  e.  Cat )
7170adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  O  e.  Cat )
725setccat 14245 . . . . . . . . . . . . 13  |-  ( U  e.  _V  ->  S  e.  Cat )
7310, 72syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  S  e.  Cat )
7473adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  S  e.  Cat )
7532, 71, 74, 48, 22, 23evlf1 14322 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  w
) ( 1st `  E
) z )  =  ( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )
)
763adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  C  e.  Cat )
77 eqid 2438 . . . . . . . . . . 11  |-  (  Hom  `  C )  =  (  Hom  `  C )
782, 17, 76, 21, 77, 23yon11 14366 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )  =  ( z (  Hom  `  C )
w ) )
7975, 78eqtrd 2470 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  w
) ( 1st `  E
) z )  =  ( z (  Hom  `  C ) w ) )
807adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  V  e.  W )
8111adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  ran  (  Homf 
`  C )  C_  U )
828adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ran  (  Homf  `  Q
)  u.  U ) 
C_  V )
832, 17, 29, 4, 5, 30, 6, 31, 26, 32, 33, 76, 80, 81, 82, 22, 23yonedalem21 14375 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  w
) ( 1st `  Z
) z )  =  ( ( ( 1st `  Y ) `  z
) ( O Nat  S
) ( ( 1st `  Y ) `  w
) ) )
8479, 83oveq12d 6102 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( ( 1st `  Y ) `
 w ) ( 1st `  E ) z ) (  Iso  `  T ) ( ( ( 1st `  Y
) `  w )
( 1st `  Z
) z ) )  =  ( ( z (  Hom  `  C
) w ) (  Iso  `  T )
( ( ( 1st `  Y ) `  z
) ( O Nat  S
) ( ( 1st `  Y ) `  w
) ) ) )
8568, 84eleqtrd 2514 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  w
) N z )  e.  ( ( z (  Hom  `  C
) w ) (  Iso  `  T )
( ( ( 1st `  Y ) `  z
) ( O Nat  S
) ( ( 1st `  Y ) `  w
) ) ) )
869adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  U  C_  V )
87 eqid 2438 . . . . . . . . . . . . 13  |-  ( Base `  S )  =  (
Base `  S )
88 relfunc 14064 . . . . . . . . . . . . . 14  |-  Rel  ( O  Func  S )
89 1st2ndbr 6399 . . . . . . . . . . . . . 14  |-  ( ( Rel  ( O  Func  S )  /\  ( ( 1st `  Y ) `
 w )  e.  ( O  Func  S
) )  ->  ( 1st `  ( ( 1st `  Y ) `  w
) ) ( O 
Func  S ) ( 2nd `  ( ( 1st `  Y
) `  w )
) )
9088, 22, 89sylancr 646 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( 1st `  (
( 1st `  Y
) `  w )
) ( O  Func  S ) ( 2nd `  (
( 1st `  Y
) `  w )
) )
9148, 87, 90funcf1 14068 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( 1st `  (
( 1st `  Y
) `  w )
) : B --> ( Base `  S ) )
9291, 23ffvelrnd 5874 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )  e.  ( Base `  S
) )
935, 10setcbas 14238 . . . . . . . . . . . 12  |-  ( ph  ->  U  =  ( Base `  S ) )
9493adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  U  =  ( Base `  S ) )
9592, 94eleqtrrd 2515 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )  e.  U )
9678, 95eqeltrrd 2513 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( z (  Hom  `  C ) w )  e.  U )
9786, 96sseldd 3351 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( z (  Hom  `  C ) w )  e.  V )
98 eqid 2438 . . . . . . . . . 10  |-  (  Homf  `  Q )  =  (  Homf 
`  Q )
99 eqid 2438 . . . . . . . . . . 11  |-  ( O Nat 
S )  =  ( O Nat  S )
1006, 99fuchom 14163 . . . . . . . . . 10  |-  ( O Nat 
S )  =  (  Hom  `  Q )
10120, 23ffvelrnd 5874 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  Y
) `  z )  e.  ( O  Func  S
) )
10298, 18, 100, 101, 22homfval 13923 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  z
) (  Homf  `  Q ) ( ( 1st `  Y
) `  w )
)  =  ( ( ( 1st `  Y
) `  z )
( O Nat  S ) ( ( 1st `  Y
) `  w )
) )
1038unssad 3526 . . . . . . . . . . 11  |-  ( ph  ->  ran  (  Homf  `  Q ) 
C_  V )
104103adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  ran  (  Homf 
`  Q )  C_  V )
10598, 18homffn 13924 . . . . . . . . . . . 12  |-  (  Homf  `  Q )  Fn  (
( O  Func  S
)  X.  ( O 
Func  S ) )
106105a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
(  Homf 
`  Q )  Fn  ( ( O  Func  S )  X.  ( O 
Func  S ) ) )
107 fnovrn 6224 . . . . . . . . . . 11  |-  ( ( (  Homf 
`  Q )  Fn  ( ( O  Func  S )  X.  ( O 
Func  S ) )  /\  ( ( 1st `  Y
) `  z )  e.  ( O  Func  S
)  /\  ( ( 1st `  Y ) `  w )  e.  ( O  Func  S )
)  ->  ( (
( 1st `  Y
) `  z )
(  Homf 
`  Q ) ( ( 1st `  Y
) `  w )
)  e.  ran  (  Homf  `  Q ) )
108106, 101, 22, 107syl3anc 1185 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  z
) (  Homf  `  Q ) ( ( 1st `  Y
) `  w )
)  e.  ran  (  Homf  `  Q ) )
109104, 108sseldd 3351 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  z
) (  Homf  `  Q ) ( ( 1st `  Y
) `  w )
)  e.  V )
110102, 109eqeltrrd 2513 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  z
) ( O Nat  S
) ( ( 1st `  Y ) `  w
) )  e.  V
)
11130, 80, 97, 110, 51setciso 14251 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( ( 1st `  Y ) `
 w ) N z )  e.  ( ( z (  Hom  `  C ) w ) (  Iso  `  T
) ( ( ( 1st `  Y ) `
 z ) ( O Nat  S ) ( ( 1st `  Y
) `  w )
) )  <->  ( (
( 1st `  Y
) `  w ) N z ) : ( z (  Hom  `  C ) w ) -1-1-onto-> ( ( ( 1st `  Y
) `  z )
( O Nat  S ) ( ( 1st `  Y
) `  w )
) ) )
11285, 111mpbid 203 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  w
) N z ) : ( z (  Hom  `  C )
w ) -1-1-onto-> ( ( ( 1st `  Y ) `  z
) ( O Nat  S
) ( ( 1st `  Y ) `  w
) ) )
11376adantr 453 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  C  e.  Cat )
114113adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  ->  C  e.  Cat )
11523adantr 453 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  z  e.  B )
116115adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  ->  z  e.  B )
117 simpr 449 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  ->  y  e.  B )
1182, 17, 114, 116, 77, 117yon11 14366 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  ->  (
( 1st `  (
( 1st `  Y
) `  z )
) `  y )  =  ( y (  Hom  `  C )
z ) )
119118eqcomd 2443 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  ->  (
y (  Hom  `  C
) z )  =  ( ( 1st `  (
( 1st `  Y
) `  z )
) `  y )
)
120114adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  /\  g  e.  ( y (  Hom  `  C ) z ) )  ->  C  e.  Cat )
12121ad3antrrr 712 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  /\  g  e.  ( y (  Hom  `  C ) z ) )  ->  w  e.  B )
122116adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  /\  g  e.  ( y (  Hom  `  C ) z ) )  ->  z  e.  B )
123 eqid 2438 . . . . . . . . . . . . . . 15  |-  (comp `  C )  =  (comp `  C )
124117adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  /\  g  e.  ( y (  Hom  `  C ) z ) )  ->  y  e.  B )
125 simpr 449 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  /\  g  e.  ( y (  Hom  `  C ) z ) )  ->  g  e.  ( y (  Hom  `  C ) z ) )
126 simpllr 737 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  /\  g  e.  ( y (  Hom  `  C ) z ) )  ->  h  e.  ( z (  Hom  `  C ) w ) )
1272, 17, 120, 121, 77, 122, 123, 124, 125, 126yon12 14367 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  /\  g  e.  ( y (  Hom  `  C ) z ) )  ->  ( (
( z ( 2nd `  ( ( 1st `  Y
) `  w )
) y ) `  g ) `  h
)  =  ( h ( <. y ,  z
>. (comp `  C )
w ) g ) )
1282, 17, 120, 122, 77, 121, 123, 124, 126, 125yon2 14368 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  /\  g  e.  ( y (  Hom  `  C ) z ) )  ->  ( (
( ( z ( 2nd `  Y ) w ) `  h
) `  y ) `  g )  =  ( h ( <. y ,  z >. (comp `  C ) w ) g ) )
129127, 128eqtr4d 2473 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  /\  g  e.  ( y (  Hom  `  C ) z ) )  ->  ( (
( z ( 2nd `  ( ( 1st `  Y
) `  w )
) y ) `  g ) `  h
)  =  ( ( ( ( z ( 2nd `  Y ) w ) `  h
) `  y ) `  g ) )
130119, 129mpteq12dva 4289 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  ->  (
g  e.  ( y (  Hom  `  C
) z )  |->  ( ( ( z ( 2nd `  ( ( 1st `  Y ) `
 w ) ) y ) `  g
) `  h )
)  =  ( g  e.  ( ( 1st `  ( ( 1st `  Y
) `  z )
) `  y )  |->  ( ( ( ( z ( 2nd `  Y
) w ) `  h ) `  y
) `  g )
) )
13116adantr 453 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
13217, 77, 100, 131, 23, 21funcf2 14070 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( z ( 2nd `  Y ) w ) : ( z (  Hom  `  C )
w ) --> ( ( ( 1st `  Y
) `  z )
( O Nat  S ) ( ( 1st `  Y
) `  w )
) )
133132ffvelrnda 5873 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  ( (
z ( 2nd `  Y
) w ) `  h )  e.  ( ( ( 1st `  Y
) `  z )
( O Nat  S ) ( ( 1st `  Y
) `  w )
) )
13499, 133nat1st2nd 14153 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  ( (
z ( 2nd `  Y
) w ) `  h )  e.  (
<. ( 1st `  (
( 1st `  Y
) `  z )
) ,  ( 2nd `  ( ( 1st `  Y
) `  z )
) >. ( O Nat  S
) <. ( 1st `  (
( 1st `  Y
) `  w )
) ,  ( 2nd `  ( ( 1st `  Y
) `  w )
) >. ) )
135134adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  ->  (
( z ( 2nd `  Y ) w ) `
 h )  e.  ( <. ( 1st `  (
( 1st `  Y
) `  z )
) ,  ( 2nd `  ( ( 1st `  Y
) `  z )
) >. ( O Nat  S
) <. ( 1st `  (
( 1st `  Y
) `  w )
) ,  ( 2nd `  ( ( 1st `  Y
) `  w )
) >. ) )
136 eqid 2438 . . . . . . . . . . . . . . 15  |-  (  Hom  `  S )  =  (  Hom  `  S )
13799, 135, 48, 136, 117natcl 14155 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  ->  (
( ( z ( 2nd `  Y ) w ) `  h
) `  y )  e.  ( ( ( 1st `  ( ( 1st `  Y
) `  z )
) `  y )
(  Hom  `  S ) ( ( 1st `  (
( 1st `  Y
) `  w )
) `  y )
) )
13810adantr 453 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  U  e.  _V )
139138ad2antrr 708 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  ->  U  e.  _V )
14019ad2antrr 708 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  ( 1st `  Y ) : B --> ( O  Func  S ) )
141140, 115ffvelrnd 5874 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  ( ( 1st `  Y ) `  z )  e.  ( O  Func  S )
)
142 1st2ndbr 6399 . . . . . . . . . . . . . . . . . . 19  |-  ( ( Rel  ( O  Func  S )  /\  ( ( 1st `  Y ) `
 z )  e.  ( O  Func  S
) )  ->  ( 1st `  ( ( 1st `  Y ) `  z
) ) ( O 
Func  S ) ( 2nd `  ( ( 1st `  Y
) `  z )
) )
14388, 141, 142sylancr 646 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  ( 1st `  ( ( 1st `  Y
) `  z )
) ( O  Func  S ) ( 2nd `  (
( 1st `  Y
) `  z )
) )
14448, 87, 143funcf1 14068 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  ( 1st `  ( ( 1st `  Y
) `  z )
) : B --> ( Base `  S ) )
145144ffvelrnda 5873 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  ->  (
( 1st `  (
( 1st `  Y
) `  z )
) `  y )  e.  ( Base `  S
) )
14694ad2antrr 708 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  ->  U  =  ( Base `  S
) )
147145, 146eleqtrrd 2515 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  ->  (
( 1st `  (
( 1st `  Y
) `  z )
) `  y )  e.  U )
14891adantr 453 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  ( 1st `  ( ( 1st `  Y
) `  w )
) : B --> ( Base `  S ) )
149148ffvelrnda 5873 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  ->  (
( 1st `  (
( 1st `  Y
) `  w )
) `  y )  e.  ( Base `  S
) )
150149, 146eleqtrrd 2515 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  ->  (
( 1st `  (
( 1st `  Y
) `  w )
) `  y )  e.  U )
1515, 139, 136, 147, 150elsetchom 14241 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  ->  (
( ( ( z ( 2nd `  Y
) w ) `  h ) `  y
)  e.  ( ( ( 1st `  (
( 1st `  Y
) `  z )
) `  y )
(  Hom  `  S ) ( ( 1st `  (
( 1st `  Y
) `  w )
) `  y )
)  <->  ( ( ( z ( 2nd `  Y
) w ) `  h ) `  y
) : ( ( 1st `  ( ( 1st `  Y ) `
 z ) ) `
 y ) --> ( ( 1st `  (
( 1st `  Y
) `  w )
) `  y )
) )
152137, 151mpbid 203 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  ->  (
( ( z ( 2nd `  Y ) w ) `  h
) `  y ) : ( ( 1st `  ( ( 1st `  Y
) `  z )
) `  y ) --> ( ( 1st `  (
( 1st `  Y
) `  w )
) `  y )
)
153152feqmptd 5782 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  ->  (
( ( z ( 2nd `  Y ) w ) `  h
) `  y )  =  ( g  e.  ( ( 1st `  (
( 1st `  Y
) `  z )
) `  y )  |->  ( ( ( ( z ( 2nd `  Y
) w ) `  h ) `  y
) `  g )
) )
154130, 153eqtr4d 2473 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z (  Hom  `  C ) w ) )  /\  y  e.  B )  ->  (
g  e.  ( y (  Hom  `  C
) z )  |->  ( ( ( z ( 2nd `  ( ( 1st `  Y ) `
 w ) ) y ) `  g
) `  h )
)  =  ( ( ( z ( 2nd `  Y ) w ) `
 h ) `  y ) )
155154mpteq2dva 4298 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C )
z )  |->  ( ( ( z ( 2nd `  ( ( 1st `  Y
) `  w )
) y ) `  g ) `  h
) ) )  =  ( y  e.  B  |->  ( ( ( z ( 2nd `  Y
) w ) `  h ) `  y
) ) )
15680adantr 453 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  V  e.  W )
15781adantr 453 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  ran  (  Homf  `  C )  C_  U
)
15882adantr 453 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  ( ran  (  Homf 
`  Q )  u.  U )  C_  V
)
15922adantr 453 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  ( ( 1st `  Y ) `  w )  e.  ( O  Func  S )
)
16078eleq2d 2505 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( h  e.  ( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )  <->  h  e.  ( z (  Hom  `  C )
w ) ) )
161160biimpar 473 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  h  e.  ( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )
)
1622, 17, 29, 4, 5, 30, 6, 31, 26, 32, 33, 113, 156, 157, 158, 159, 115, 44, 161yonedalem4a 14377 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  ( (
( ( 1st `  Y
) `  w ) N z ) `  h )  =  ( y  e.  B  |->  ( g  e.  ( y (  Hom  `  C
) z )  |->  ( ( ( z ( 2nd `  ( ( 1st `  Y ) `
 w ) ) y ) `  g
) `  h )
) ) )
16399, 134, 48natfn 14156 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  ( (
z ( 2nd `  Y
) w ) `  h )  Fn  B
)
164 dffn5 5775 . . . . . . . . . . 11  |-  ( ( ( z ( 2nd `  Y ) w ) `
 h )  Fn  B  <->  ( ( z ( 2nd `  Y
) w ) `  h )  =  ( y  e.  B  |->  ( ( ( z ( 2nd `  Y ) w ) `  h
) `  y )
) )
165163, 164sylib 190 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  ( (
z ( 2nd `  Y
) w ) `  h )  =  ( y  e.  B  |->  ( ( ( z ( 2nd `  Y ) w ) `  h
) `  y )
) )
166155, 162, 1653eqtr4d 2480 . . . . . . . . 9  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z (  Hom  `  C ) w ) )  ->  ( (
( ( 1st `  Y
) `  w ) N z ) `  h )  =  ( ( z ( 2nd `  Y ) w ) `
 h ) )
167166mpteq2dva 4298 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( h  e.  ( z (  Hom  `  C
) w )  |->  ( ( ( ( 1st `  Y ) `  w
) N z ) `
 h ) )  =  ( h  e.  ( z (  Hom  `  C ) w ) 
|->  ( ( z ( 2nd `  Y ) w ) `  h
) ) )
168 f1of 5677 . . . . . . . . . 10  |-  ( ( ( ( 1st `  Y
) `  w ) N z ) : ( z (  Hom  `  C ) w ) -1-1-onto-> ( ( ( 1st `  Y
) `  z )
( O Nat  S ) ( ( 1st `  Y
) `  w )
)  ->  ( (
( 1st `  Y
) `  w ) N z ) : ( z (  Hom  `  C ) w ) --> ( ( ( 1st `  Y ) `  z
) ( O Nat  S
) ( ( 1st `  Y ) `  w
) ) )
169112, 168syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  w
) N z ) : ( z (  Hom  `  C )
w ) --> ( ( ( 1st `  Y
) `  z )
( O Nat  S ) ( ( 1st `  Y
) `  w )
) )
170169feqmptd 5782 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  w
) N z )  =  ( h  e.  ( z (  Hom  `  C ) w ) 
|->  ( ( ( ( 1st `  Y ) `
 w ) N z ) `  h
) ) )
171132feqmptd 5782 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( z ( 2nd `  Y ) w )  =  ( h  e.  ( z (  Hom  `  C ) w ) 
|->  ( ( z ( 2nd `  Y ) w ) `  h
) ) )
172167, 170, 1713eqtr4d 2480 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  w
) N z )  =  ( z ( 2nd `  Y ) w ) )
173 f1oeq1 5668 . . . . . . 7  |-  ( ( ( ( 1st `  Y
) `  w ) N z )  =  ( z ( 2nd `  Y ) w )  ->  ( ( ( ( 1st `  Y
) `  w ) N z ) : ( z (  Hom  `  C ) w ) -1-1-onto-> ( ( ( 1st `  Y
) `  z )
( O Nat  S ) ( ( 1st `  Y
) `  w )
)  <->  ( z ( 2nd `  Y ) w ) : ( z (  Hom  `  C
) w ) -1-1-onto-> ( ( ( 1st `  Y
) `  z )
( O Nat  S ) ( ( 1st `  Y
) `  w )
) ) )
174172, 173syl 16 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( ( 1st `  Y ) `
 w ) N z ) : ( z (  Hom  `  C
) w ) -1-1-onto-> ( ( ( 1st `  Y
) `  z )
( O Nat  S ) ( ( 1st `  Y
) `  w )
)  <->  ( z ( 2nd `  Y ) w ) : ( z (  Hom  `  C
) w ) -1-1-onto-> ( ( ( 1st `  Y
) `  z )
( O Nat  S ) ( ( 1st `  Y
) `  w )
) ) )
175112, 174mpbid 203 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( z ( 2nd `  Y ) w ) : ( z (  Hom  `  C )
w ) -1-1-onto-> ( ( ( 1st `  Y ) `  z
) ( O Nat  S
) ( ( 1st `  Y ) `  w
) ) )
176175ralrimivva 2800 . . . 4  |-  ( ph  ->  A. z  e.  B  A. w  e.  B  ( z ( 2nd `  Y ) w ) : ( z (  Hom  `  C )
w ) -1-1-onto-> ( ( ( 1st `  Y ) `  z
) ( O Nat  S
) ( ( 1st `  Y ) `  w
) ) )
17717, 77, 100isffth2 14118 . . . 4  |-  ( ( 1st `  Y ) ( ( C Full  Q
)  i^i  ( C Faith  Q ) ) ( 2nd `  Y )  <->  ( ( 1st `  Y ) ( C  Func  Q )
( 2nd `  Y
)  /\  A. z  e.  B  A. w  e.  B  ( z
( 2nd `  Y
) w ) : ( z (  Hom  `  C ) w ) -1-1-onto-> ( ( ( 1st `  Y
) `  z )
( O Nat  S ) ( ( 1st `  Y
) `  w )
) ) )
17816, 176, 177sylanbrc 647 . . 3  |-  ( ph  ->  ( 1st `  Y
) ( ( C Full 
Q )  i^i  ( C Faith  Q ) ) ( 2nd `  Y ) )
179 df-br 4216 . . 3  |-  ( ( 1st `  Y ) ( ( C Full  Q
)  i^i  ( C Faith  Q ) ) ( 2nd `  Y )  <->  <. ( 1st `  Y ) ,  ( 2nd `  Y )
>.  e.  ( ( C Full 
Q )  i^i  ( C Faith  Q ) ) )
180178, 179sylib 190 . 2  |-  ( ph  -> 
<. ( 1st `  Y
) ,  ( 2nd `  Y ) >.  e.  ( ( C Full  Q )  i^i  ( C Faith  Q
) ) )
18114, 180eqeltrd 2512 1  |-  ( ph  ->  Y  e.  ( ( C Full  Q )  i^i  ( C Faith  Q ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    u. cun 3320    i^i cin 3321    C_ wss 3322   <.cop 3819   class class class wbr 4215    e. cmpt 4269    X. cxp 4879   ran crn 4882   Rel wrel 4886    Fn wfn 5452   -->wf 5453   -1-1-onto->wf1o 5456   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1stc1st 6350   2ndc2nd 6351  tpos ctpos 6481   Basecbs 13474    Hom chom 13545  compcco 13546   Catccat 13894   Idccid 13895    Homf chomf 13896  oppCatcoppc 13942  Invcinv 13976    Iso ciso 13977    Func cfunc 14056    o.func ccofu 14058   Full cful 14104   Faith cfth 14105   Nat cnat 14143   FuncCat cfuc 14144   SetCatcsetc 14235    X.c cxpc 14270    1stF c1stf 14271    2ndF c2ndf 14272   ⟨,⟩F cprf 14273   evalF cevlf 14311  HomFchof 14350  Yoncyon 14351
This theorem is referenced by:  yonffth  14386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-tpos 6482  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-fz 11049  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-hom 13558  df-cco 13559  df-cat 13898  df-cid 13899  df-homf 13900  df-comf 13901  df-oppc 13943  df-sect 13978  df-inv 13979  df-iso 13980  df-ssc 14015  df-resc 14016  df-subc 14017  df-func 14060  df-cofu 14062  df-full 14106  df-fth 14107  df-nat 14145  df-fuc 14146  df-setc 14236  df-xpc 14274  df-1stf 14275  df-2ndf 14276  df-prf 14277  df-evlf 14315  df-curf 14316  df-hof 14352  df-yon 14353
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