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Theorem yonffthlem 15409
Description: Lemma for yonffth 15411. (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 15089 . . 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 3682 . . . . 5  |-  ( ph  ->  U  C_  V )
107, 9ssexd 4594 . . . 4  |-  ( ph  ->  U  e.  _V )
11 yoneda.u . . . 4  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
122, 3, 4, 5, 6, 10, 11yoncl 15389 . . 3  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
13 1st2nd 6830 . . 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 6833 . . . . 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 15187 . . . . . . . . . . . . 13  |-  ( O 
Func  S )  =  (
Base `  Q )
1917, 18, 16funcf1 15093 . . . . . . . . . . . 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 6022 . . . . . . . . . 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 5031 . . . . . . . . . 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 15187 . . . . . . . . . . . . 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 15399 . . . . . . . . . . . . . . . . 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 15090 . . . . . . . . . . . . . . . 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 15197 . . . . . . . . . . . . 13  |-  ( ph  ->  R  e.  Cat )
4134simprd 463 . . . . . . . . . . . . 13  |-  ( ph  ->  E  e.  ( ( Q  X.c  O )  Func  T
) )
42 eqid 2467 . . . . . . . . . . . . 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 15408 . . . . . . . . . . . . 13  |-  ( ph  ->  M ( Z I E ) N )
4627, 28, 40, 35, 41, 42, 45inviso2 15022 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  ( E (  Iso  `  R
) Z ) )
47 eqid 2467 . . . . . . . . . . . . . 14  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
484, 17oppcbas 14974 . . . . . . . . . . . . . 14  |-  B  =  ( Base `  O
)
4947, 18, 48xpcbas 15305 . . . . . . . . . . . . 13  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
50 eqid 2467 . . . . . . . . . . . . 13  |-  ( ( Q  X.c  O ) Nat  T )  =  ( ( Q  X.c  O ) Nat  T )
51 eqid 2467 . . . . . . . . . . . . 13  |-  (  Iso  `  T )  =  (  Iso  `  T )
5226, 49, 50, 41, 35, 42, 51fuciso 15202 . . . . . . . . . . . 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 5866 . . . . . . . . . . . 12  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( N `  v
)  =  ( N `
 <. ( ( 1st `  Y ) `  w
) ,  z >.
) )
57 df-ov 6287 . . . . . . . . . . . 12  |-  ( ( ( 1st `  Y
) `  w ) N z )  =  ( N `  <. ( ( 1st `  Y
) `  w ) ,  z >. )
5856, 57syl6eqr 2526 . . . . . . . . . . 11  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( N `  v
)  =  ( ( ( 1st `  Y
) `  w ) N z ) )
59 fveq2 5866 . . . . . . . . . . . . 13  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( ( 1st `  E
) `  v )  =  ( ( 1st `  E ) `  <. ( ( 1st `  Y
) `  w ) ,  z >. )
)
60 df-ov 6287 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  Y
) `  w )
( 1st `  E
) z )  =  ( ( 1st `  E
) `  <. ( ( 1st `  Y ) `
 w ) ,  z >. )
6159, 60syl6eqr 2526 . . . . . . . . . . . 12  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( ( 1st `  E
) `  v )  =  ( ( ( 1st `  Y ) `
 w ) ( 1st `  E ) z ) )
62 fveq2 5866 . . . . . . . . . . . . 13  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( ( 1st `  Z
) `  v )  =  ( ( 1st `  Z ) `  <. ( ( 1st `  Y
) `  w ) ,  z >. )
)
63 df-ov 6287 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  Y
) `  w )
( 1st `  Z
) z )  =  ( ( 1st `  Z
) `  <. ( ( 1st `  Y ) `
 w ) ,  z >. )
6462, 63syl6eqr 2526 . . . . . . . . . . . 12  |-  ( v  =  <. ( ( 1st `  Y ) `  w
) ,  z >.  ->  ( ( 1st `  Z
) `  v )  =  ( ( ( 1st `  Y ) `
 w ) ( 1st `  Z ) z ) )
6561, 64oveq12d 6302 . . . . . . . . . . 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 2549 . . . . . . . . . 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 3210 . . . . . . . . 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 14978 . . . . . . . . . . . . 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 15270 . . . . . . . . . . . . 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 15347 . . . . . . . . . 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 2467 . . . . . . . . . . 11  |-  ( Hom  `  C )  =  ( Hom  `  C )
782, 17, 76, 21, 77, 23yon11 15391 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )  =  ( z ( Hom  `  C )
w ) )
7975, 78eqtrd 2508 . . . . . . . . 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 15400 . . . . . . . . 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 6302 . . . . . . . 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 2557 . . . . . . 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 2467 . . . . . . . . . . . . 13  |-  ( Base `  S )  =  (
Base `  S )
88 relfunc 15089 . . . . . . . . . . . . . 14  |-  Rel  ( O  Func  S )
89 1st2ndbr 6833 . . . . . . . . . . . . . 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 15093 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( 1st `  (
( 1st `  Y
) `  w )
) : B --> ( Base `  S ) )
9291, 23ffvelrnd 6022 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )  e.  ( Base `  S
) )
935, 10setcbas 15263 . . . . . . . . . . . 12  |-  ( ph  ->  U  =  ( Base `  S ) )
9493adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  ->  U  =  ( Base `  S ) )
9592, 94eleqtrrd 2558 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  (
( 1st `  Y
) `  w )
) `  z )  e.  U )
9678, 95eqeltrrd 2556 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( z ( Hom  `  C ) w )  e.  U )
9786, 96sseldd 3505 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( z ( Hom  `  C ) w )  e.  V )
98 eqid 2467 . . . . . . . . . 10  |-  ( Hom f  `  Q )  =  ( Hom f  `  Q )
99 eqid 2467 . . . . . . . . . . 11  |-  ( O Nat 
S )  =  ( O Nat  S )
1006, 99fuchom 15188 . . . . . . . . . 10  |-  ( O Nat 
S )  =  ( Hom  `  Q )
10120, 23ffvelrnd 6022 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( 1st `  Y
) `  z )  e.  ( O  Func  S
) )
10298, 18, 100, 101, 22homfval 14948 . . . . . . . . 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 3681 . . . . . . . . . . 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 14949 . . . . . . . . . . . 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 6434 . . . . . . . . . . 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 1228 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  z
) ( Hom f  `  Q ) ( ( 1st `  Y
) `  w )
)  e.  ran  ( Hom f  `  Q ) )
109104, 108sseldd 3505 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  z
) ( Hom f  `  Q ) ( ( 1st `  Y
) `  w )
)  e.  V )
110102, 109eqeltrrd 2556 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  z
) ( O Nat  S
) ( ( 1st `  Y ) `  w
) )  e.  V
)
11130, 80, 97, 110, 51setciso 15276 . . . . . . 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 15391 . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . 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 15392 . . . . . . . . . . . . . 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 15393 . . . . . . . . . . . . . 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 2511 . . . . . . . . . . . . 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 4524 . . . . . . . . . . . 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 15095 . . . . . . . . . . . . . . . . . 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 6021 . . . . . . . . . . . . . . . . 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 15178 . . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . 15  |-  ( Hom  `  S )  =  ( Hom  `  S )
13799, 135, 48, 136, 117natcl 15180 . . . . . . . . . . . . . 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 6022 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ( ( 1st `  Y ) `  z )  e.  ( O  Func  S )
)
142 1st2ndbr 6833 . . . . . . . . . . . . . . . . . . 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 15093 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ( 1st `  ( ( 1st `  Y
) `  z )
) : B --> ( Base `  S ) )
145144ffvelrnda 6021 . . . . . . . . . . . . . . . 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 2558 . . . . . . . . . . . . . . 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 6021 . . . . . . . . . . . . . . . 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 2558 . . . . . . . . . . . . . . 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 15266 . . . . . . . . . . . . . 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 5920 . . . . . . . . . . . 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 2511 . . . . . . . . . . 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 4533 . . . . . . . . . 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 2537 . . . . . . . . . . . 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 15402 . . . . . . . . . 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 15181 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  B  /\  w  e.  B )
)  /\  h  e.  ( z ( Hom  `  C ) w ) )  ->  ( (
z ( 2nd `  Y
) w ) `  h )  Fn  B
)
164 dffn5 5913 . . . . . . . . . . 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 2518 . . . . . . . . 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 4533 . . . . . . . 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 5816 . . . . . . . . . 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 5920 . . . . . . . 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 5920 . . . . . . . 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 2518 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  w  e.  B ) )  -> 
( ( ( 1st `  Y ) `  w
) N z )  =  ( z ( 2nd `  Y ) w ) )
173 f1oeq1 5807 . . . . . . 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 2885 . . . 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 15143 . . . 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 4448 . . 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 2555 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 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    u. cun 3474    i^i cin 3475    C_ wss 3476   <.cop 4033   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   ran crn 5000   Rel wrel 5004    Fn wfn 5583   -->wf 5584   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   1stc1st 6782   2ndc2nd 6783  tpos ctpos 6954   Basecbs 14490   Hom chom 14566  compcco 14567   Catccat 14919   Idccid 14920   Hom f chomf 14921  oppCatcoppc 14967  Invcinv 15001    Iso ciso 15002    Func cfunc 15081    o.func ccofu 15083   Full cful 15129   Faith cfth 15130   Nat cnat 15168   FuncCat cfuc 15169   SetCatcsetc 15260    X.c cxpc 15295    1stF c1stf 15296    2ndF c2ndf 15297   ⟨,⟩F cprf 15298   evalF cevlf 15336  HomFchof 15375  Yoncyon 15376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-fz 11673  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-hom 14579  df-cco 14580  df-cat 14923  df-cid 14924  df-homf 14925  df-comf 14926  df-oppc 14968  df-sect 15003  df-inv 15004  df-iso 15005  df-ssc 15040  df-resc 15041  df-subc 15042  df-func 15085  df-cofu 15087  df-full 15131  df-fth 15132  df-nat 15170  df-fuc 15171  df-setc 15261  df-xpc 15299  df-1stf 15300  df-2ndf 15301  df-prf 15302  df-evlf 15340  df-curf 15341  df-hof 15377  df-yon 15378
This theorem is referenced by:  yonffth  15411
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