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

Theorem yonffthlem 15092
Description: Lemma for yonffth 15094. (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 14772 . . 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 3534 . . . . 5  |-  ( ph  ->  U  C_  V )
107, 9ssexd 4439 . . . 4  |-  ( ph  ->  U  e.  _V )
11 yoneda.u . . . 4  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
122, 3, 4, 5, 6, 10, 11yoncl 15072 . . 3  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
13 1st2nd 6620 . . 3  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  Y  =  <. ( 1st `  Y
) ,  ( 2nd `  Y ) >. )
141, 12, 13sylancr 663 . 2  |-  ( ph  ->  Y  =  <. ( 1st `  Y ) ,  ( 2nd `  Y
) >. )
15 1st2ndbr 6623 . . . . 5  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  ( 1st `  Y ) ( C  Func  Q )
( 2nd `  Y
) )
161, 12, 15sylancr 663 . . . 4  |-  ( ph  ->  ( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
17 yoneda.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  C
)
186fucbas 14870 . . . . . . . . . . . . 13  |-  ( O 
Func  S )  =  (
Base `  Q )
1917, 18, 16funcf1 14776 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st `  Y
) : B --> ( O 
Func  S ) )
2019adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( 1st `  Y
) : B --> ( O 
Func  S ) )
21 simprr 756 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  w  e.  B )
2220, 21ffvelrnd 5844 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  Y
) `  w )  e.  ( O  Func  S
) )
23 simprl 755 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
z  e.  B )
24 opelxpi 4871 . . . . . . . . . 10  |-  ( ( ( ( 1st `  Y
) `  w )  e.  ( O  Func  S
)  /\  z  e.  B )  ->  <. (
( 1st `  Y
) `  w ) ,  z >.  e.  ( ( O  Func  S
)  X.  B ) )
2522, 23, 24syl2anc 661 . . . . . . . . 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 14870 . . . . . . . . . . . . 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 15082 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( Z  e.  ( ( Q  X.c  O ) 
Func  T )  /\  E  e.  ( ( Q  X.c  O
)  Func  T )
) )
3534simpld 459 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  Z  e.  ( ( Q  X.c  O )  Func  T
) )
36 funcrcl 14773 . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( Q  X.c  O )  e.  Cat )
3937simprd 463 . . . . . . . . . . . . . 14  |-  ( ph  ->  T  e.  Cat )
4026, 38, 39fuccat 14880 . . . . . . . . . . . . 13  |-  ( ph  ->  R  e.  Cat )
4134simprd 463 . . . . . . . . . . . . 13  |-  ( ph  ->  E  e.  ( ( Q  X.c  O )  Func  T
) )
42 eqid 2443 . . . . . . . . . . . . 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 15091 . . . . . . . . . . . . 13  |-  ( ph  ->  M ( Z I E ) N )
4627, 28, 40, 35, 41, 42, 45inviso2 14705 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  ( E (  Iso  `  R
) Z ) )
47 eqid 2443 . . . . . . . . . . . . . 14  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
484, 17oppcbas 14657 . . . . . . . . . . . . . 14  |-  B  =  ( Base `  O
)
4947, 18, 48xpcbas 14988 . . . . . . . . . . . . 13  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
50 eqid 2443 . . . . . . . . . . . . 13  |-  ( ( Q  X.c  O ) Nat  T )  =  ( ( Q  X.c  O ) Nat  T )
51 eqid 2443 . . . . . . . . . . . . 13  |-  (  Iso  `  T )  =  (  Iso  `  T )
5226, 49, 50, 41, 35, 42, 51fuciso 14885 . . . . . . . . . . . 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 463 . . . . . . . . . 10  |-  ( ph  ->  A. v  e.  ( ( O  Func  S
)  X.  B ) ( N `  v
)  e.  ( ( ( 1st `  E
) `  v )
(  Iso  `  T ) ( ( 1st `  Z
) `  v )
) )
5554adantr 465 . . . . . . . . 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 5691 . . . . . . . . . . . 12  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( N `  v
)  =  ( N `
 <. ( ( 1st `  Y ) `  w
) ,  z >.
) )
57 df-ov 6094 . . . . . . . . . . . 12  |-  ( ( ( 1st `  Y
) `  w ) N z )  =  ( N `  <. ( ( 1st `  Y
) `  w ) ,  z >. )
5856, 57syl6eqr 2493 . . . . . . . . . . 11  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( N `  v
)  =  ( ( ( 1st `  Y
) `  w ) N z ) )
59 fveq2 5691 . . . . . . . . . . . . 13  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( ( 1st `  E
) `  v )  =  ( ( 1st `  E ) `  <. ( ( 1st `  Y
) `  w ) ,  z >. )
)
60 df-ov 6094 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  Y
) `  w )
( 1st `  E
) z )  =  ( ( 1st `  E
) `  <. ( ( 1st `  Y ) `
 w ) ,  z >. )
6159, 60syl6eqr 2493 . . . . . . . . . . . 12  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( ( 1st `  E
) `  v )  =  ( ( ( 1st `  Y ) `
 w ) ( 1st `  E ) z ) )
62 fveq2 5691 . . . . . . . . . . . . 13  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( ( 1st `  Z
) `  v )  =  ( ( 1st `  Z ) `  <. ( ( 1st `  Y
) `  w ) ,  z >. )
)
63 df-ov 6094 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  Y
) `  w )
( 1st `  Z
) z )  =  ( ( 1st `  Z
) `  <. ( ( 1st `  Y ) `
 w ) ,  z >. )
6462, 63syl6eqr 2493 . . . . . . . . . . . 12  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( ( 1st `  Z
) `  v )  =  ( ( ( 1st `  Y ) `
 w ) ( 1st `  Z ) z ) )
6561, 64oveq12d 6109 . . . . . . . . . . 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 2511 . . . . . . . . . 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 3069 . . . . . . . . 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 14661 . . . . . . . . . . . . 13  |-  ( C  e.  Cat  ->  O  e.  Cat )
703, 69syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  O  e.  Cat )
7170adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  O  e.  Cat )
725setccat 14953 . . . . . . . . . . . . 13  |-  ( U  e.  _V  ->  S  e.  Cat )
7310, 72syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  S  e.  Cat )
7473adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  S  e.  Cat )
7532, 71, 74, 48, 22, 23evlf1 15030 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  w
) ( 1st `  E
) z )  =  ( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )
)
763adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  C  e.  Cat )
77 eqid 2443 . . . . . . . . . . 11  |-  ( Hom  `  C )  =  ( Hom  `  C )
782, 17, 76, 21, 77, 23yon11 15074 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )  =  ( z ( Hom  `  C )
w ) )
7975, 78eqtrd 2475 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  w
) ( 1st `  E
) z )  =  ( z ( Hom  `  C ) w ) )
807adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  V  e.  W )
8111adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  ran  ( Hom f  `  C )  C_  U )
828adantr 465 . . . . . . . . . 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 15083 . . . . . . . . 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 6109 . . . . . . . 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 2519 . . . . . . 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 465 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  U  C_  V )
87 eqid 2443 . . . . . . . . . . . . 13  |-  ( Base `  S )  =  (
Base `  S )
88 relfunc 14772 . . . . . . . . . . . . . 14  |-  Rel  ( O  Func  S )
89 1st2ndbr 6623 . . . . . . . . . . . . . 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 663 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( 1st `  (
( 1st `  Y
) `  w )
) ( O  Func  S ) ( 2nd `  (
( 1st `  Y
) `  w )
) )
9148, 87, 90funcf1 14776 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( 1st `  (
( 1st `  Y
) `  w )
) : B --> ( Base `  S ) )
9291, 23ffvelrnd 5844 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )  e.  ( Base `  S
) )
935, 10setcbas 14946 . . . . . . . . . . . 12  |-  ( ph  ->  U  =  ( Base `  S ) )
9493adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  U  =  ( Base `  S ) )
9592, 94eleqtrrd 2520 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )  e.  U )
9678, 95eqeltrrd 2518 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( z ( Hom  `  C ) w )  e.  U )
9786, 96sseldd 3357 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( z ( Hom  `  C ) w )  e.  V )
98 eqid 2443 . . . . . . . . . 10  |-  ( Hom f  `  Q )  =  ( Hom f  `  Q )
99 eqid 2443 . . . . . . . . . . 11  |-  ( O Nat 
S )  =  ( O Nat  S )
1006, 99fuchom 14871 . . . . . . . . . 10  |-  ( O Nat 
S )  =  ( Hom  `  Q )
10120, 23ffvelrnd 5844 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  Y
) `  z )  e.  ( O  Func  S
) )
10298, 18, 100, 101, 22homfval 14631 . . . . . . . . 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 3533 . . . . . . . . . . 11  |-  ( ph  ->  ran  ( Hom f  `  Q ) 
C_  V )
104103adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  ran  ( Hom f  `  Q )  C_  V )
10598, 18homffn 14632 . . . . . . . . . . . 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 6238 . . . . . . . . . . 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 1218 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  z
) ( Hom f  `  Q ) ( ( 1st `  Y
) `  w )
)  e.  ran  ( Hom f  `  Q ) )
109104, 108sseldd 3357 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  z
) ( Hom f  `  Q ) ( ( 1st `  Y
) `  w )
)  e.  V )
110102, 109eqeltrrd 2518 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  z
) ( O Nat  S
) ( ( 1st `  Y ) `  w
) )  e.  V
)
11130, 80, 97, 110, 51setciso 14959 . . . . . . 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 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  C  e.  Cat )
114113adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z ( Hom  `  C ) w ) )  /\  y  e.  B )  ->  C  e.  Cat )
11523adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  z  e.  B )
116115adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z ( Hom  `  C ) w ) )  /\  y  e.  B )  ->  z  e.  B )
117 simpr 461 . . . . . . . . . . . . . . 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 15074 . . . . . . . . . . . . . 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 2448 . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . 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 729 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . . 15  |-  (comp `  C )  =  (comp `  C )
124117adantr 465 . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . 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 15075 . . . . . . . . . . . . . 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 15076 . . . . . . . . . . . . . 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 2478 . . . . . . . . . . . . 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 4369 . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
13217, 77, 100, 131, 23, 21funcf2 14778 . . . . . . . . . . . . . . . . . 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 5843 . . . . . . . . . . . . . . . . 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 14861 . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . . 15  |-  ( Hom  `  S )  =  ( Hom  `  S )
13799, 135, 48, 136, 117natcl 14863 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  U  e.  _V )
139138ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z ( Hom  `  C ) w ) )  /\  y  e.  B )  ->  U  e.  _V )
14019ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ( 1st `  Y ) : B --> ( O  Func  S ) )
141140, 115ffvelrnd 5844 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ( ( 1st `  Y ) `  z )  e.  ( O  Func  S )
)
142 1st2ndbr 6623 . . . . . . . . . . . . . . . . . . 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 663 . . . . . . . . . . . . . . . . . 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 14776 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ( 1st `  ( ( 1st `  Y
) `  z )
) : B --> ( Base `  S ) )
145144ffvelrnda 5843 . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( z  e.  B  /\  w  e.  B
) )  /\  h  e.  ( z ( Hom  `  C ) w ) )  /\  y  e.  B )  ->  U  =  ( Base `  S
) )
147145, 146eleqtrrd 2520 . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ( 1st `  ( ( 1st `  Y
) `  w )
) : B --> ( Base `  S ) )
149148ffvelrnda 5843 . . . . . . . . . . . . . . . 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 2520 . . . . . . . . . . . . . . 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 14949 . . . . . . . . . . . . . 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 5744 . . . . . . . . . . . 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 2478 . . . . . . . . . . 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 4378 . . . . . . . . . 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 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  V  e.  W )
15781adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ran  ( Hom f  `  C )  C_  U
)
15882adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ( ran  ( Hom f  `  Q )  u.  U
)  C_  V )
15922adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ( ( 1st `  Y ) `  w )  e.  ( O  Func  S )
)
16078eleq2d 2510 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( h  e.  ( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )  <->  h  e.  ( z ( Hom  `  C )
w ) ) )
161160biimpar 485 . . . . . . . . . . 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 15085 . . . . . . . . . 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 14864 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ( (
z ( 2nd `  Y
) w ) `  h )  Fn  B
)
164 dffn5 5737 . . . . . . . . . . 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 2485 . . . . . . . . 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 4378 . . . . . . . 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 5641 . . . . . . . . . 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 5744 . . . . . . . 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 5744 . . . . . . . 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 2485 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  w
) N z )  =  ( z ( 2nd `  Y ) w ) )
173 f1oeq1 5632 . . . . . . 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 2808 . . . 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 14826 . . . 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 664 . . 3  |-  ( ph  ->  ( 1st `  Y
) ( ( C Full 
Q )  i^i  ( C Faith  Q ) ) ( 2nd `  Y ) )
179 df-br 4293 . . 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 2517 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 1369    e. wcel 1756   A.wral 2715   _Vcvv 2972    u. cun 3326    i^i cin 3327    C_ wss 3328   <.cop 3883   class class class wbr 4292    e. cmpt 4350    X. cxp 4838   ran crn 4841   Rel wrel 4845    Fn wfn 5413   -->wf 5414   -1-1-onto->wf1o 5417   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   1stc1st 6575   2ndc2nd 6576  tpos ctpos 6744   Basecbs 14174   Hom chom 14249  compcco 14250   Catccat 14602   Idccid 14603   Hom f chomf 14604  oppCatcoppc 14650  Invcinv 14684    Iso ciso 14685    Func cfunc 14764    o.func ccofu 14766   Full cful 14812   Faith cfth 14813   Nat cnat 14851   FuncCat cfuc 14852   SetCatcsetc 14943    X.c cxpc 14978    1stF c1stf 14979    2ndF c2ndf 14980   ⟨,⟩F cprf 14981   evalF cevlf 15019  HomFchof 15058  Yoncyon 15059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-tpos 6745  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-fz 11438  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-hom 14262  df-cco 14263  df-cat 14606  df-cid 14607  df-homf 14608  df-comf 14609  df-oppc 14651  df-sect 14686  df-inv 14687  df-iso 14688  df-ssc 14723  df-resc 14724  df-subc 14725  df-func 14768  df-cofu 14770  df-full 14814  df-fth 14815  df-nat 14853  df-fuc 14854  df-setc 14944  df-xpc 14982  df-1stf 14983  df-2ndf 14984  df-prf 14985  df-evlf 15023  df-curf 15024  df-hof 15060  df-yon 15061
This theorem is referenced by:  yonffth  15094
  Copyright terms: Public domain W3C validator