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Theorem yonffthlem 15088
Description: Lemma for yonffth 15090. (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 )
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 14768 . . 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  ( Hom f  `  Q )  u.  U
)  C_  V )
98unssbd 3531 . . . . 5  |-  ( ph  ->  U  C_  V )
107, 9ssexd 4436 . . . 4  |-  ( ph  ->  U  e.  _V )
11 yoneda.u . . . 4  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
122, 3, 4, 5, 6, 10, 11yoncl 15068 . . 3  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
13 1st2nd 6619 . . 3  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  Y  =  <. ( 1st `  Y
) ,  ( 2nd `  Y ) >. )
141, 12, 13sylancr 658 . 2  |-  ( ph  ->  Y  =  <. ( 1st `  Y ) ,  ( 2nd `  Y
) >. )
15 1st2ndbr 6622 . . . . 5  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  ( 1st `  Y ) ( C  Func  Q )
( 2nd `  Y
) )
161, 12, 15sylancr 658 . . . 4  |-  ( ph  ->  ( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
17 yoneda.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  C
)
186fucbas 14866 . . . . . . . . . . . . 13  |-  ( O 
Func  S )  =  (
Base `  Q )
1917, 18, 16funcf1 14772 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st `  Y
) : B --> ( O 
Func  S ) )
2019adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( 1st `  Y
) : B --> ( O 
Func  S ) )
21 simprr 751 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  w  e.  B )
2220, 21ffvelrnd 5841 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  Y
) `  w )  e.  ( O  Func  S
) )
23 simprl 750 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
z  e.  B )
24 opelxpi 4867 . . . . . . . . . 10  |-  ( ( ( ( 1st `  Y
) `  w )  e.  ( O  Func  S
)  /\  z  e.  B )  ->  <. (
( 1st `  Y
) `  w ) ,  z >.  e.  ( ( O  Func  S
)  X.  B ) )
2522, 23, 24syl2anc 656 . . . . . . . . 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 14866 . . . . . . . . . . . . 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 15078 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( Z  e.  ( ( Q  X.c  O ) 
Func  T )  /\  E  e.  ( ( Q  X.c  O
)  Func  T )
) )
3534simpld 456 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  Z  e.  ( ( Q  X.c  O )  Func  T
) )
36 funcrcl 14769 . . . . . . . . . . . . . . . 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 456 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( Q  X.c  O )  e.  Cat )
3937simprd 460 . . . . . . . . . . . . . 14  |-  ( ph  ->  T  e.  Cat )
4026, 38, 39fuccat 14876 . . . . . . . . . . . . 13  |-  ( ph  ->  R  e.  Cat )
4134simprd 460 . . . . . . . . . . . . 13  |-  ( ph  ->  E  e.  ( ( Q  X.c  O )  Func  T
) )
42 eqid 2441 . . . . . . . . . . . . 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 15087 . . . . . . . . . . . . 13  |-  ( ph  ->  M ( Z I E ) N )
4627, 28, 40, 35, 41, 42, 45inviso2 14701 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  ( E (  Iso  `  R
) Z ) )
47 eqid 2441 . . . . . . . . . . . . . 14  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
484, 17oppcbas 14653 . . . . . . . . . . . . . 14  |-  B  =  ( Base `  O
)
4947, 18, 48xpcbas 14984 . . . . . . . . . . . . 13  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
50 eqid 2441 . . . . . . . . . . . . 13  |-  ( ( Q  X.c  O ) Nat  T )  =  ( ( Q  X.c  O ) Nat  T )
51 eqid 2441 . . . . . . . . . . . . 13  |-  (  Iso  `  T )  =  (  Iso  `  T )
5226, 49, 50, 41, 35, 42, 51fuciso 14881 . . . . . . . . . . . 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 210 . . . . . . . . . . 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 460 . . . . . . . . . 10  |-  ( ph  ->  A. v  e.  ( ( O  Func  S
)  X.  B ) ( N `  v
)  e.  ( ( ( 1st `  E
) `  v )
(  Iso  `  T ) ( ( 1st `  Z
) `  v )
) )
5554adantr 462 . . . . . . . . 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 5688 . . . . . . . . . . . 12  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( N `  v
)  =  ( N `
 <. ( ( 1st `  Y ) `  w
) ,  z >.
) )
57 df-ov 6093 . . . . . . . . . . . 12  |-  ( ( ( 1st `  Y
) `  w ) N z )  =  ( N `  <. ( ( 1st `  Y
) `  w ) ,  z >. )
5856, 57syl6eqr 2491 . . . . . . . . . . 11  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( N `  v
)  =  ( ( ( 1st `  Y
) `  w ) N z ) )
59 fveq2 5688 . . . . . . . . . . . . 13  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( ( 1st `  E
) `  v )  =  ( ( 1st `  E ) `  <. ( ( 1st `  Y
) `  w ) ,  z >. )
)
60 df-ov 6093 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  Y
) `  w )
( 1st `  E
) z )  =  ( ( 1st `  E
) `  <. ( ( 1st `  Y ) `
 w ) ,  z >. )
6159, 60syl6eqr 2491 . . . . . . . . . . . 12  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( ( 1st `  E
) `  v )  =  ( ( ( 1st `  Y ) `
 w ) ( 1st `  E ) z ) )
62 fveq2 5688 . . . . . . . . . . . . 13  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( ( 1st `  Z
) `  v )  =  ( ( 1st `  Z ) `  <. ( ( 1st `  Y
) `  w ) ,  z >. )
)
63 df-ov 6093 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  Y
) `  w )
( 1st `  Z
) z )  =  ( ( 1st `  Z
) `  <. ( ( 1st `  Y ) `
 w ) ,  z >. )
6462, 63syl6eqr 2491 . . . . . . . . . . . 12  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( ( 1st `  Z
) `  v )  =  ( ( ( 1st `  Y ) `
 w ) ( 1st `  Z ) z ) )
6561, 64oveq12d 6108 . . . . . . . . . . 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 2509 . . . . . . . . . 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 3066 . . . . . . . . 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 60 . . . . . . . 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 14657 . . . . . . . . . . . . 13  |-  ( C  e.  Cat  ->  O  e.  Cat )
703, 69syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  O  e.  Cat )
7170adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  O  e.  Cat )
725setccat 14949 . . . . . . . . . . . . 13  |-  ( U  e.  _V  ->  S  e.  Cat )
7310, 72syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  S  e.  Cat )
7473adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  S  e.  Cat )
7532, 71, 74, 48, 22, 23evlf1 15026 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  w
) ( 1st `  E
) z )  =  ( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )
)
763adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  C  e.  Cat )
77 eqid 2441 . . . . . . . . . . 11  |-  ( Hom  `  C )  =  ( Hom  `  C )
782, 17, 76, 21, 77, 23yon11 15070 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )  =  ( z ( Hom  `  C )
w ) )
7975, 78eqtrd 2473 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  w
) ( 1st `  E
) z )  =  ( z ( Hom  `  C ) w ) )
807adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  V  e.  W )
8111adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  ran  ( Hom f  `  C )  C_  U )
828adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ran  ( Hom f  `  Q
)  u.  U ) 
C_  V )
832, 17, 29, 4, 5, 30, 6, 31, 26, 32, 33, 76, 80, 81, 82, 22, 23yonedalem21 15079 . . . . . . . . 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 6108 . . . . . . . 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 2517 . . . . . . 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 462 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  U  C_  V )
87 eqid 2441 . . . . . . . . . . . . 13  |-  ( Base `  S )  =  (
Base `  S )
88 relfunc 14768 . . . . . . . . . . . . . 14  |-  Rel  ( O  Func  S )
89 1st2ndbr 6622 . . . . . . . . . . . . . 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 658 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( 1st `  (
( 1st `  Y
) `  w )
) ( O  Func  S ) ( 2nd `  (
( 1st `  Y
) `  w )
) )
9148, 87, 90funcf1 14772 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( 1st `  (
( 1st `  Y
) `  w )
) : B --> ( Base `  S ) )
9291, 23ffvelrnd 5841 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )  e.  ( Base `  S
) )
935, 10setcbas 14942 . . . . . . . . . . . 12  |-  ( ph  ->  U  =  ( Base `  S ) )
9493adantr 462 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  U  =  ( Base `  S ) )
9592, 94eleqtrrd 2518 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )  e.  U )
9678, 95eqeltrrd 2516 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( z ( Hom  `  C ) w )  e.  U )
9786, 96sseldd 3354 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( z ( Hom  `  C ) w )  e.  V )
98 eqid 2441 . . . . . . . . . 10  |-  ( Hom f  `  Q )  =  ( Hom f  `  Q )
99 eqid 2441 . . . . . . . . . . 11  |-  ( O Nat 
S )  =  ( O Nat  S )
1006, 99fuchom 14867 . . . . . . . . . 10  |-  ( O Nat 
S )  =  ( Hom  `  Q )
10120, 23ffvelrnd 5841 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  Y
) `  z )  e.  ( O  Func  S
) )
10298, 18, 100, 101, 22homfval 14627 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  z
) ( Hom f  `  Q ) ( ( 1st `  Y
) `  w )
)  =  ( ( ( 1st `  Y
) `  z )
( O Nat  S ) ( ( 1st `  Y
) `  w )
) )
1038unssad 3530 . . . . . . . . . . 11  |-  ( ph  ->  ran  ( Hom f  `  Q ) 
C_  V )
104103adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  ran  ( Hom f  `  Q )  C_  V )
10598, 18homffn 14628 . . . . . . . . . . . 12  |-  ( Hom f  `  Q )  Fn  (
( O  Func  S
)  X.  ( O 
Func  S ) )
106105a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( Hom f  `  Q )  Fn  ( ( O  Func  S )  X.  ( O 
Func  S ) ) )
107 fnovrn 6237 . . . . . . . . . . 11  |-  ( ( ( Hom f  `  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 )
( Hom f  `  Q ) ( ( 1st `  Y
) `  w )
)  e.  ran  ( Hom f  `  Q ) )
108106, 101, 22, 107syl3anc 1213 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  z
) ( Hom f  `  Q ) ( ( 1st `  Y
) `  w )
)  e.  ran  ( Hom f  `  Q ) )
109104, 108sseldd 3354 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  z
) ( Hom f  `  Q ) ( ( 1st `  Y
) `  w )
)  e.  V )
110102, 109eqeltrrd 2516 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  z
) ( O Nat  S
) ( ( 1st `  Y ) `  w
) )  e.  V
)
11130, 80, 97, 110, 51setciso 14955 . . . . . . 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 210 . . . . . 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 462 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  C  e.  Cat )
114113adantr 462 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z ( Hom  `  C ) w ) )  /\  y  e.  B )  ->  C  e.  Cat )
11523adantr 462 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  z  e.  B )
116115adantr 462 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z ( Hom  `  C ) w ) )  /\  y  e.  B )  ->  z  e.  B )
117 simpr 458 . . . . . . . . . . . . . . 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 15070 . . . . . . . . . . . . . 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 2446 . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . 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 2441 . . . . . . . . . . . . . . 15  |-  (comp `  C )  =  (comp `  C )
124117adantr 462 . . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . . 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 753 . . . . . . . . . . . . . . 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 15071 . . . . . . . . . . . . . 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 15072 . . . . . . . . . . . . . 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 2476 . . . . . . . . . . . . 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 4366 . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
13217, 77, 100, 131, 23, 21funcf2 14774 . . . . . . . . . . . . . . . . . 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 5840 . . . . . . . . . . . . . . . . 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 14857 . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . 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 2441 . . . . . . . . . . . . . . 15  |-  ( Hom  `  S )  =  ( Hom  `  S )
13799, 135, 48, 136, 117natcl 14859 . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  U  e.  _V )
139138ad2antrr 720 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z ( Hom  `  C ) w ) )  /\  y  e.  B )  ->  U  e.  _V )
14019ad2antrr 720 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ( 1st `  Y ) : B --> ( O  Func  S ) )
141140, 115ffvelrnd 5841 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ( ( 1st `  Y ) `  z )  e.  ( O  Func  S )
)
142 1st2ndbr 6622 . . . . . . . . . . . . . . . . . . 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 658 . . . . . . . . . . . . . . . . . 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 14772 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ( 1st `  ( ( 1st `  Y
) `  z )
) : B --> ( Base `  S ) )
145144ffvelrnda 5840 . . . . . . . . . . . . . . . 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 720 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z ( Hom  `  C ) w ) )  /\  y  e.  B )  ->  U  =  ( Base `  S
) )
147145, 146eleqtrrd 2518 . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ( 1st `  ( ( 1st `  Y
) `  w )
) : B --> ( Base `  S ) )
149148ffvelrnda 5840 . . . . . . . . . . . . . . . 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 2518 . . . . . . . . . . . . . . 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 14945 . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . 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 5741 . . . . . . . . . . . 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 2476 . . . . . . . . . . 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 4375 . . . . . . . . . 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 462 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  V  e.  W )
15781adantr 462 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ran  ( Hom f  `  C )  C_  U
)
15882adantr 462 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ( ran  ( Hom f  `  Q )  u.  U
)  C_  V )
15922adantr 462 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ( ( 1st `  Y ) `  w )  e.  ( O  Func  S )
)
16078eleq2d 2508 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( h  e.  ( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )  <->  h  e.  ( z ( Hom  `  C )
w ) ) )
161160biimpar 482 . . . . . . . . . . 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 15081 . . . . . . . . . 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 14860 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ( (
z ( 2nd `  Y
) w ) `  h )  Fn  B
)
164 dffn5 5734 . . . . . . . . . . 11  |-  ( ( ( z ( 2nd `  Y ) w ) `
 h )  Fn  B  <->  ( ( z ( 2nd `  Y
) w ) `  h )  =  ( y  e.  B  |->  ( ( ( z ( 2nd `  Y ) w ) `  h
) `  y )
) )
165163, 164sylib 196 . . . . . . . . . 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 2483 . . . . . . . . 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 4375 . . . . . . . 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 5638 . . . . . . . . . 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 5741 . . . . . . . 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 5741 . . . . . . . 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 2483 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  w
) N z )  =  ( z ( 2nd `  Y ) w ) )
173 f1oeq1 5629 . . . . . . 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 210 . . . . 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 2806 . . . 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 14822 . . . 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 659 . . 3  |-  ( ph  ->  ( 1st `  Y
) ( ( C Full 
Q )  i^i  ( C Faith  Q ) ) ( 2nd `  Y ) )
179 df-br 4290 . . 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 196 . 2  |-  ( ph  -> 
<. ( 1st `  Y
) ,  ( 2nd `  Y ) >.  e.  ( ( C Full  Q )  i^i  ( C Faith  Q
) ) )
18114, 180eqeltrd 2515 1  |-  ( ph  ->  Y  e.  ( ( C Full  Q )  i^i  ( C Faith  Q ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970    u. cun 3323    i^i cin 3324    C_ wss 3325   <.cop 3880   class class class wbr 4289    e. cmpt 4347    X. cxp 4834   ran crn 4837   Rel wrel 4841    Fn wfn 5410   -->wf 5411   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   1stc1st 6574   2ndc2nd 6575  tpos ctpos 6743   Basecbs 14170   Hom chom 14245  compcco 14246   Catccat 14598   Idccid 14599   Hom f chomf 14600  oppCatcoppc 14646  Invcinv 14680    Iso ciso 14681    Func cfunc 14760    o.func ccofu 14762   Full cful 14808   Faith cfth 14809   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 2263  df-mo 2264  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-pm 7213  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-ress 14177  df-hom 14258  df-cco 14259  df-cat 14602  df-cid 14603  df-homf 14604  df-comf 14605  df-oppc 14647  df-sect 14682  df-inv 14683  df-iso 14684  df-ssc 14719  df-resc 14720  df-subc 14721  df-func 14764  df-cofu 14766  df-full 14810  df-fth 14811  df-nat 14849  df-fuc 14850  df-setc 14940  df-xpc 14978  df-1stf 14979  df-2ndf 14980  df-prf 14981  df-evlf 15019  df-curf 15020  df-hof 15056  df-yon 15057
This theorem is referenced by:  yonffth  15090
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