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Theorem yonedalem22 15084
Description: Lemma for yoneda 15089. (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 )
yonedalem21.f  |-  ( ph  ->  F  e.  ( O 
Func  S ) )
yonedalem21.x  |-  ( ph  ->  X  e.  B )
yonedalem22.g  |-  ( ph  ->  G  e.  ( O 
Func  S ) )
yonedalem22.p  |-  ( ph  ->  P  e.  B )
yonedalem22.a  |-  ( ph  ->  A  e.  ( F ( O Nat  S ) G ) )
yonedalem22.k  |-  ( ph  ->  K  e.  ( P ( Hom  `  C
) X ) )
Assertion
Ref Expression
yonedalem22  |-  ( ph  ->  ( A ( <. F ,  X >. ( 2nd `  Z )
<. G ,  P >. ) K )  =  ( ( ( P ( 2nd `  Y ) X ) `  K
) ( <. (
( 1st `  Y
) `  X ) ,  F >. ( 2nd `  H
) <. ( ( 1st `  Y ) `  P
) ,  G >. ) A ) )

Proof of Theorem yonedalem22
StepHypRef Expression
1 yoneda.z . . . . . . 7  |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )
21fveq2i 5691 . . . . . 6  |-  ( 2nd `  Z )  =  ( 2nd `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) )
32oveqi 6103 . . . . 5  |-  ( <. F ,  X >. ( 2nd `  Z )
<. G ,  P >. )  =  ( <. F ,  X >. ( 2nd `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) ) <. G ,  P >. )
43oveqi 6103 . . . 4  |-  ( A ( <. F ,  X >. ( 2nd `  Z
) <. G ,  P >. ) K )  =  ( A ( <. F ,  X >. ( 2nd `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) ) <. G ,  P >. ) K )
5 df-ov 6093 . . . 4  |-  ( A ( <. F ,  X >. ( 2nd `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) ) <. G ,  P >. ) K )  =  ( ( <. F ,  X >. ( 2nd `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) ) <. G ,  P >. ) `  <. A ,  K >. )
64, 5eqtri 2461 . . 3  |-  ( A ( <. F ,  X >. ( 2nd `  Z
) <. G ,  P >. ) K )  =  ( ( <. F ,  X >. ( 2nd `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) ) <. G ,  P >. ) `  <. A ,  K >. )
7 eqid 2441 . . . . 5  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
8 yoneda.q . . . . . 6  |-  Q  =  ( O FuncCat  S )
98fucbas 14866 . . . . 5  |-  ( O 
Func  S )  =  (
Base `  Q )
10 yoneda.o . . . . . 6  |-  O  =  (oppCat `  C )
11 yoneda.b . . . . . 6  |-  B  =  ( Base `  C
)
1210, 11oppcbas 14653 . . . . 5  |-  B  =  ( Base `  O
)
137, 9, 12xpcbas 14984 . . . 4  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
14 eqid 2441 . . . . 5  |-  ( (
<. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
)  =  ( (
<. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
)
15 eqid 2441 . . . . 5  |-  ( (oppCat `  Q )  X.c  Q )  =  ( (oppCat `  Q )  X.c  Q )
16 yoneda.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
1710oppccat 14657 . . . . . . . . 9  |-  ( C  e.  Cat  ->  O  e.  Cat )
1816, 17syl 16 . . . . . . . 8  |-  ( ph  ->  O  e.  Cat )
19 yoneda.w . . . . . . . . . 10  |-  ( ph  ->  V  e.  W )
20 yoneda.v . . . . . . . . . . 11  |-  ( ph  ->  ( ran  ( Hom f  `  Q )  u.  U
)  C_  V )
2120unssbd 3531 . . . . . . . . . 10  |-  ( ph  ->  U  C_  V )
2219, 21ssexd 4436 . . . . . . . . 9  |-  ( ph  ->  U  e.  _V )
23 yoneda.s . . . . . . . . . 10  |-  S  =  ( SetCat `  U )
2423setccat 14949 . . . . . . . . 9  |-  ( U  e.  _V  ->  S  e.  Cat )
2522, 24syl 16 . . . . . . . 8  |-  ( ph  ->  S  e.  Cat )
268, 18, 25fuccat 14876 . . . . . . 7  |-  ( ph  ->  Q  e.  Cat )
27 eqid 2441 . . . . . . 7  |-  ( Q  2ndF  O )  =  ( Q  2ndF  O )
287, 26, 18, 272ndfcl 15004 . . . . . 6  |-  ( ph  ->  ( Q  2ndF  O )  e.  ( ( Q  X.c  O
)  Func  O )
)
29 eqid 2441 . . . . . . . 8  |-  (oppCat `  Q )  =  (oppCat `  Q )
30 relfunc 14768 . . . . . . . . 9  |-  Rel  ( C  Func  Q )
31 yoneda.y . . . . . . . . . 10  |-  Y  =  (Yon `  C )
32 yoneda.u . . . . . . . . . 10  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
3331, 16, 10, 23, 8, 22, 32yoncl 15068 . . . . . . . . 9  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
34 1st2ndbr 6622 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  ( 1st `  Y ) ( C  Func  Q )
( 2nd `  Y
) )
3530, 33, 34sylancr 658 . . . . . . . 8  |-  ( ph  ->  ( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
3610, 29, 35funcoppc 14781 . . . . . . 7  |-  ( ph  ->  ( 1st `  Y
) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y ) )
37 df-br 4290 . . . . . . 7  |-  ( ( 1st `  Y ) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y )  <->  <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  e.  ( O  Func  (oppCat `  Q ) ) )
3836, 37sylib 196 . . . . . 6  |-  ( ph  -> 
<. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  e.  ( O  Func  (oppCat `  Q
) ) )
3928, 38cofucl 14794 . . . . 5  |-  ( ph  ->  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) )  e.  ( ( Q  X.c  O ) 
Func  (oppCat `  Q )
) )
40 eqid 2441 . . . . . 6  |-  ( Q  1stF  O )  =  ( Q  1stF  O )
417, 26, 18, 401stfcl 15003 . . . . 5  |-  ( ph  ->  ( Q  1stF  O )  e.  ( ( Q  X.c  O
)  Func  Q )
)
4214, 15, 39, 41prfcl 15009 . . . 4  |-  ( ph  ->  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) )  e.  ( ( Q  X.c  O
)  Func  ( (oppCat `  Q )  X.c  Q ) ) )
43 yoneda.h . . . . 5  |-  H  =  (HomF
`  Q )
44 yoneda.t . . . . 5  |-  T  =  ( SetCat `  V )
4520unssad 3530 . . . . 5  |-  ( ph  ->  ran  ( Hom f  `  Q ) 
C_  V )
4643, 29, 44, 26, 19, 45hofcl 15065 . . . 4  |-  ( ph  ->  H  e.  ( ( (oppCat `  Q )  X.c  Q )  Func  T
) )
47 yonedalem21.f . . . . 5  |-  ( ph  ->  F  e.  ( O 
Func  S ) )
48 yonedalem21.x . . . . 5  |-  ( ph  ->  X  e.  B )
49 opelxpi 4867 . . . . 5  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  <. F ,  X >.  e.  ( ( O  Func  S )  X.  B ) )
5047, 48, 49syl2anc 656 . . . 4  |-  ( ph  -> 
<. F ,  X >.  e.  ( ( O  Func  S )  X.  B ) )
51 yonedalem22.g . . . . 5  |-  ( ph  ->  G  e.  ( O 
Func  S ) )
52 yonedalem22.p . . . . 5  |-  ( ph  ->  P  e.  B )
53 opelxpi 4867 . . . . 5  |-  ( ( G  e.  ( O 
Func  S )  /\  P  e.  B )  ->  <. G ,  P >.  e.  ( ( O  Func  S )  X.  B ) )
5451, 52, 53syl2anc 656 . . . 4  |-  ( ph  -> 
<. G ,  P >.  e.  ( ( O  Func  S )  X.  B ) )
55 eqid 2441 . . . 4  |-  ( Hom  `  ( Q  X.c  O ) )  =  ( Hom  `  ( Q  X.c  O ) )
56 yonedalem22.a . . . . . 6  |-  ( ph  ->  A  e.  ( F ( O Nat  S ) G ) )
57 yonedalem22.k . . . . . . 7  |-  ( ph  ->  K  e.  ( P ( Hom  `  C
) X ) )
58 eqid 2441 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
5958, 10oppchom 14650 . . . . . . 7  |-  ( X ( Hom  `  O
) P )  =  ( P ( Hom  `  C ) X )
6057, 59syl6eleqr 2532 . . . . . 6  |-  ( ph  ->  K  e.  ( X ( Hom  `  O
) P ) )
61 opelxpi 4867 . . . . . 6  |-  ( ( A  e.  ( F ( O Nat  S ) G )  /\  K  e.  ( X ( Hom  `  O ) P ) )  ->  <. A ,  K >.  e.  ( ( F ( O Nat  S
) G )  X.  ( X ( Hom  `  O ) P ) ) )
6256, 60, 61syl2anc 656 . . . . 5  |-  ( ph  -> 
<. A ,  K >.  e.  ( ( F ( O Nat  S ) G )  X.  ( X ( Hom  `  O
) P ) ) )
63 eqid 2441 . . . . . . 7  |-  ( O Nat 
S )  =  ( O Nat  S )
648, 63fuchom 14867 . . . . . 6  |-  ( O Nat 
S )  =  ( Hom  `  Q )
65 eqid 2441 . . . . . 6  |-  ( Hom  `  O )  =  ( Hom  `  O )
667, 9, 12, 64, 65, 47, 48, 51, 52, 55xpchom2 14992 . . . . 5  |-  ( ph  ->  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. )  =  ( ( F ( O Nat  S ) G )  X.  ( X ( Hom  `  O
) P ) ) )
6762, 66eleqtrrd 2518 . . . 4  |-  ( ph  -> 
<. A ,  K >.  e.  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) )
6813, 42, 46, 50, 54, 55, 67cofu2 14792 . . 3  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) ) <. G ,  P >. ) `  <. A ,  K >. )  =  ( ( ( ( 1st `  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) `
 <. F ,  X >. ) ( 2nd `  H
) ( ( 1st `  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) `
 <. G ,  P >. ) ) `  (
( <. F ,  X >. ( 2nd `  (
( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) <. G ,  P >. ) `  <. A ,  K >. )
) )
696, 68syl5eq 2485 . 2  |-  ( ph  ->  ( A ( <. F ,  X >. ( 2nd `  Z )
<. G ,  P >. ) K )  =  ( ( ( ( 1st `  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) `
 <. F ,  X >. ) ( 2nd `  H
) ( ( 1st `  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) `
 <. G ,  P >. ) ) `  (
( <. F ,  X >. ( 2nd `  (
( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) <. G ,  P >. ) `  <. A ,  K >. )
) )
7014, 13, 55, 39, 41, 50prf1 15006 . . . . . 6  |-  ( ph  ->  ( ( 1st `  (
( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) `  <. F ,  X >. )  =  <. ( ( 1st `  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ) `  <. F ,  X >. ) ,  ( ( 1st `  ( Q  1stF  O )
) `  <. F ,  X >. ) >. )
7113, 28, 38, 50cofu1 14790 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ) `  <. F ,  X >. )  =  ( ( 1st `  <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >. ) `  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) ) )
72 fvex 5698 . . . . . . . . . . 11  |-  ( 1st `  Y )  e.  _V
73 fvex 5698 . . . . . . . . . . . 12  |-  ( 2nd `  Y )  e.  _V
7473tposex 6778 . . . . . . . . . . 11  |- tpos  ( 2nd `  Y )  e.  _V
7572, 74op1st 6584 . . . . . . . . . 10  |-  ( 1st `  <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >. )  =  ( 1st `  Y
)
7675a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. )  =  ( 1st `  Y ) )
777, 13, 55, 26, 18, 27, 502ndf1 15001 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  ( 2nd `  <. F ,  X >. )
)
78 op2ndg 6589 . . . . . . . . . . 11  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  ( 2nd `  <. F ,  X >. )  =  X )
7947, 48, 78syl2anc 656 . . . . . . . . . 10  |-  ( ph  ->  ( 2nd `  <. F ,  X >. )  =  X )
8077, 79eqtrd 2473 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  X )
8176, 80fveq12d 5694 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. ) `  (
( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) )  =  ( ( 1st `  Y ) `
 X ) )
8271, 81eqtrd 2473 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ) `  <. F ,  X >. )  =  ( ( 1st `  Y ) `  X
) )
837, 13, 55, 26, 18, 40, 501stf1 14998 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  ( 1st `  <. F ,  X >. )
)
84 op1stg 6588 . . . . . . . . 9  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  ( 1st `  <. F ,  X >. )  =  F )
8547, 48, 84syl2anc 656 . . . . . . . 8  |-  ( ph  ->  ( 1st `  <. F ,  X >. )  =  F )
8683, 85eqtrd 2473 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  F )
8782, 86opeq12d 4064 . . . . . 6  |-  ( ph  -> 
<. ( ( 1st `  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ) `  <. F ,  X >. ) ,  ( ( 1st `  ( Q  1stF  O )
) `  <. F ,  X >. ) >.  =  <. ( ( 1st `  Y
) `  X ) ,  F >. )
8870, 87eqtrd 2473 . . . . 5  |-  ( ph  ->  ( ( 1st `  (
( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) `  <. F ,  X >. )  =  <. ( ( 1st `  Y ) `  X
) ,  F >. )
8914, 13, 55, 39, 41, 54prf1 15006 . . . . . 6  |-  ( ph  ->  ( ( 1st `  (
( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) `  <. G ,  P >. )  =  <. ( ( 1st `  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ) `  <. G ,  P >. ) ,  ( ( 1st `  ( Q  1stF  O )
) `  <. G ,  P >. ) >. )
9013, 28, 38, 54cofu1 14790 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ) `  <. G ,  P >. )  =  ( ( 1st `  <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >. ) `  ( ( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. ) ) )
917, 13, 55, 26, 18, 27, 542ndf1 15001 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. )  =  ( 2nd `  <. G ,  P >. )
)
92 op2ndg 6589 . . . . . . . . . . 11  |-  ( ( G  e.  ( O 
Func  S )  /\  P  e.  B )  ->  ( 2nd `  <. G ,  P >. )  =  P )
9351, 52, 92syl2anc 656 . . . . . . . . . 10  |-  ( ph  ->  ( 2nd `  <. G ,  P >. )  =  P )
9491, 93eqtrd 2473 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. )  =  P )
9576, 94fveq12d 5694 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. ) `  (
( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. ) )  =  ( ( 1st `  Y ) `
 P ) )
9690, 95eqtrd 2473 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ) `  <. G ,  P >. )  =  ( ( 1st `  Y ) `  P
) )
977, 13, 55, 26, 18, 40, 541stf1 14998 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. G ,  P >. )  =  ( 1st `  <. G ,  P >. )
)
98 op1stg 6588 . . . . . . . . 9  |-  ( ( G  e.  ( O 
Func  S )  /\  P  e.  B )  ->  ( 1st `  <. G ,  P >. )  =  G )
9951, 52, 98syl2anc 656 . . . . . . . 8  |-  ( ph  ->  ( 1st `  <. G ,  P >. )  =  G )
10097, 99eqtrd 2473 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. G ,  P >. )  =  G )
10196, 100opeq12d 4064 . . . . . 6  |-  ( ph  -> 
<. ( ( 1st `  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ) `  <. G ,  P >. ) ,  ( ( 1st `  ( Q  1stF  O )
) `  <. G ,  P >. ) >.  =  <. ( ( 1st `  Y
) `  P ) ,  G >. )
10289, 101eqtrd 2473 . . . . 5  |-  ( ph  ->  ( ( 1st `  (
( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) `  <. G ,  P >. )  =  <. ( ( 1st `  Y ) `  P
) ,  G >. )
10388, 102oveq12d 6108 . . . 4  |-  ( ph  ->  ( ( ( 1st `  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) `
 <. F ,  X >. ) ( 2nd `  H
) ( ( 1st `  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) `
 <. G ,  P >. ) )  =  (
<. ( ( 1st `  Y
) `  X ) ,  F >. ( 2nd `  H
) <. ( ( 1st `  Y ) `  P
) ,  G >. ) )
10414, 13, 55, 39, 41, 50, 54, 67prf2 15008 . . . . 5  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  (
( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) <. G ,  P >. ) `  <. A ,  K >. )  =  <. ( ( <. F ,  X >. ( 2nd `  ( <.
( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ) <. G ,  P >. ) `
 <. A ,  K >. ) ,  ( (
<. F ,  X >. ( 2nd `  ( Q  1stF  O ) ) <. G ,  P >. ) `
 <. A ,  K >. ) >. )
10513, 28, 38, 50, 54, 55, 67cofu2 14792 . . . . . . 7  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ) <. G ,  P >. ) `
 <. A ,  K >. )  =  ( ( ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) ( 2nd `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. ) ( ( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. ) ) `  ( (
<. F ,  X >. ( 2nd `  ( Q  2ndF  O ) ) <. G ,  P >. ) `
 <. A ,  K >. ) ) )
10672, 74op2nd 6585 . . . . . . . . . . 11  |-  ( 2nd `  <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >. )  = tpos  ( 2nd `  Y
)
107106oveqi 6103 . . . . . . . . . 10  |-  ( ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) ( 2nd `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. ) ( ( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. ) )  =  ( ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )tpos  ( 2nd `  Y
) ( ( 1st `  ( Q  2ndF  O )
) `  <. G ,  P >. ) )
108 ovtpos 6759 . . . . . . . . . 10  |-  ( ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )tpos  ( 2nd `  Y
) ( ( 1st `  ( Q  2ndF  O )
) `  <. G ,  P >. ) )  =  ( ( ( 1st `  ( Q  2ndF  O )
) `  <. G ,  P >. ) ( 2nd `  Y ) ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) )
109107, 108eqtri 2461 . . . . . . . . 9  |-  ( ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) ( 2nd `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. ) ( ( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. ) )  =  ( ( ( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. ) ( 2nd `  Y
) ( ( 1st `  ( Q  2ndF  O )
) `  <. F ,  X >. ) )
11094, 80oveq12d 6108 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 1st `  ( Q  2ndF  O )
) `  <. G ,  P >. ) ( 2nd `  Y ) ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) )  =  ( P ( 2nd `  Y
) X ) )
111109, 110syl5eq 2485 . . . . . . . 8  |-  ( ph  ->  ( ( ( 1st `  ( Q  2ndF  O )
) `  <. F ,  X >. ) ( 2nd `  <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >. )
( ( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. ) )  =  ( P ( 2nd `  Y
) X ) )
1127, 13, 55, 26, 18, 27, 50, 542ndf2 15002 . . . . . . . . . 10  |-  ( ph  ->  ( <. F ,  X >. ( 2nd `  ( Q  2ndF  O ) ) <. G ,  P >. )  =  ( 2nd  |`  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) ) )
113112fveq1d 5690 . . . . . . . . 9  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  ( Q  2ndF  O ) ) <. G ,  P >. ) `
 <. A ,  K >. )  =  ( ( 2nd  |`  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O
) ) <. G ,  P >. ) ) `  <. A ,  K >. ) )
114 fvres 5701 . . . . . . . . . 10  |-  ( <. A ,  K >.  e.  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. )  ->  ( ( 2nd  |`  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) ) `  <. A ,  K >. )  =  ( 2nd `  <. A ,  K >. ) )
11567, 114syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( 2nd  |`  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) ) `  <. A ,  K >. )  =  ( 2nd `  <. A ,  K >. ) )
116 op2ndg 6589 . . . . . . . . . 10  |-  ( ( A  e.  ( F ( O Nat  S ) G )  /\  K  e.  ( P ( Hom  `  C ) X ) )  ->  ( 2nd ` 
<. A ,  K >. )  =  K )
11756, 57, 116syl2anc 656 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  <. A ,  K >. )  =  K )
118113, 115, 1173eqtrd 2477 . . . . . . . 8  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  ( Q  2ndF  O ) ) <. G ,  P >. ) `
 <. A ,  K >. )  =  K )
119111, 118fveq12d 5694 . . . . . . 7  |-  ( ph  ->  ( ( ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) ( 2nd `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. ) ( ( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. ) ) `  ( (
<. F ,  X >. ( 2nd `  ( Q  2ndF  O ) ) <. G ,  P >. ) `
 <. A ,  K >. ) )  =  ( ( P ( 2nd `  Y ) X ) `
 K ) )
120105, 119eqtrd 2473 . . . . . 6  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ) <. G ,  P >. ) `
 <. A ,  K >. )  =  ( ( P ( 2nd `  Y
) X ) `  K ) )
1217, 13, 55, 26, 18, 40, 50, 541stf2 14999 . . . . . . . 8  |-  ( ph  ->  ( <. F ,  X >. ( 2nd `  ( Q  1stF  O ) ) <. G ,  P >. )  =  ( 1st  |`  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) ) )
122121fveq1d 5690 . . . . . . 7  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  ( Q  1stF  O ) ) <. G ,  P >. ) `
 <. A ,  K >. )  =  ( ( 1st  |`  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O
) ) <. G ,  P >. ) ) `  <. A ,  K >. ) )
123 fvres 5701 . . . . . . . 8  |-  ( <. A ,  K >.  e.  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. )  ->  ( ( 1st  |`  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) ) `  <. A ,  K >. )  =  ( 1st `  <. A ,  K >. ) )
12467, 123syl 16 . . . . . . 7  |-  ( ph  ->  ( ( 1st  |`  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) ) `  <. A ,  K >. )  =  ( 1st `  <. A ,  K >. ) )
125 op1stg 6588 . . . . . . . 8  |-  ( ( A  e.  ( F ( O Nat  S ) G )  /\  K  e.  ( P ( Hom  `  C ) X ) )  ->  ( 1st ` 
<. A ,  K >. )  =  A )
12656, 57, 125syl2anc 656 . . . . . . 7  |-  ( ph  ->  ( 1st `  <. A ,  K >. )  =  A )
127122, 124, 1263eqtrd 2477 . . . . . 6  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  ( Q  1stF  O ) ) <. G ,  P >. ) `
 <. A ,  K >. )  =  A )
128120, 127opeq12d 4064 . . . . 5  |-  ( ph  -> 
<. ( ( <. F ,  X >. ( 2nd `  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ) <. G ,  P >. ) `
 <. A ,  K >. ) ,  ( (
<. F ,  X >. ( 2nd `  ( Q  1stF  O ) ) <. G ,  P >. ) `
 <. A ,  K >. ) >.  =  <. ( ( P ( 2nd `  Y ) X ) `
 K ) ,  A >. )
129104, 128eqtrd 2473 . . . 4  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  (
( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) <. G ,  P >. ) `  <. A ,  K >. )  =  <. ( ( P ( 2nd `  Y
) X ) `  K ) ,  A >. )
130103, 129fveq12d 5694 . . 3  |-  ( ph  ->  ( ( ( ( 1st `  ( (
<. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) `  <. F ,  X >. )
( 2nd `  H
) ( ( 1st `  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) `
 <. G ,  P >. ) ) `  (
( <. F ,  X >. ( 2nd `  (
( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) <. G ,  P >. ) `  <. A ,  K >. )
)  =  ( (
<. ( ( 1st `  Y
) `  X ) ,  F >. ( 2nd `  H
) <. ( ( 1st `  Y ) `  P
) ,  G >. ) `
 <. ( ( P ( 2nd `  Y
) X ) `  K ) ,  A >. ) )
131 df-ov 6093 . . 3  |-  ( ( ( P ( 2nd `  Y ) X ) `
 K ) (
<. ( ( 1st `  Y
) `  X ) ,  F >. ( 2nd `  H
) <. ( ( 1st `  Y ) `  P
) ,  G >. ) A )  =  ( ( <. ( ( 1st `  Y ) `  X
) ,  F >. ( 2nd `  H )
<. ( ( 1st `  Y
) `  P ) ,  G >. ) `  <. ( ( P ( 2nd `  Y ) X ) `
 K ) ,  A >. )
132130, 131syl6eqr 2491 . 2  |-  ( ph  ->  ( ( ( ( 1st `  ( (
<. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) `  <. F ,  X >. )
( 2nd `  H
) ( ( 1st `  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) `
 <. G ,  P >. ) ) `  (
( <. F ,  X >. ( 2nd `  (
( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) <. G ,  P >. ) `  <. A ,  K >. )
)  =  ( ( ( P ( 2nd `  Y ) X ) `
 K ) (
<. ( ( 1st `  Y
) `  X ) ,  F >. ( 2nd `  H
) <. ( ( 1st `  Y ) `  P
) ,  G >. ) A ) )
13369, 132eqtrd 2473 1  |-  ( ph  ->  ( A ( <. F ,  X >. ( 2nd `  Z )
<. G ,  P >. ) K )  =  ( ( ( P ( 2nd `  Y ) X ) `  K
) ( <. (
( 1st `  Y
) `  X ) ,  F >. ( 2nd `  H
) <. ( ( 1st `  Y ) `  P
) ,  G >. ) A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1761   _Vcvv 2970    u. cun 3323    C_ wss 3325   <.cop 3880   class class class wbr 4289    X. cxp 4834   ran crn 4837    |` cres 4838   Rel wrel 4841   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575  tpos ctpos 6743   Basecbs 14170   Hom chom 14245   Catccat 14598   Idccid 14599   Hom f chomf 14600  oppCatcoppc 14646    Func cfunc 14760    o.func ccofu 14762   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 2261  df-mo 2262  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-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-hom 14258  df-cco 14259  df-cat 14602  df-cid 14603  df-homf 14604  df-comf 14605  df-oppc 14647  df-func 14764  df-cofu 14766  df-nat 14849  df-fuc 14850  df-setc 14940  df-xpc 14978  df-1stf 14979  df-2ndf 14980  df-prf 14981  df-curf 15020  df-hof 15056  df-yon 15057
This theorem is referenced by:  yonedalem3b  15085
  Copyright terms: Public domain W3C validator