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Theorem yonedalem22 16107
Description: Lemma for yoneda 16112. (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 5875 . . . . . 6  |-  ( 2nd `  Z )  =  ( 2nd `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) )
32oveqi 6309 . . . . 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 6309 . . . 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 6299 . . . 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 2449 . . 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 2420 . . . . 5  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
8 yoneda.q . . . . . 6  |-  Q  =  ( O FuncCat  S )
98fucbas 15809 . . . . 5  |-  ( O 
Func  S )  =  (
Base `  Q )
10 yoneda.o . . . . . 6  |-  O  =  (oppCat `  C )
11 yoneda.b . . . . . 6  |-  B  =  ( Base `  C
)
1210, 11oppcbas 15567 . . . . 5  |-  B  =  ( Base `  O
)
137, 9, 12xpcbas 16007 . . . 4  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
14 eqid 2420 . . . . 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 2420 . . . . 5  |-  ( (oppCat `  Q )  X.c  Q )  =  ( (oppCat `  Q )  X.c  Q )
16 yoneda.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
1710oppccat 15571 . . . . . . . . 9  |-  ( C  e.  Cat  ->  O  e.  Cat )
1816, 17syl 17 . . . . . . . 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 3641 . . . . . . . . . 10  |-  ( ph  ->  U  C_  V )
2219, 21ssexd 4563 . . . . . . . . 9  |-  ( ph  ->  U  e.  _V )
23 yoneda.s . . . . . . . . . 10  |-  S  =  ( SetCat `  U )
2423setccat 15924 . . . . . . . . 9  |-  ( U  e.  _V  ->  S  e.  Cat )
2522, 24syl 17 . . . . . . . 8  |-  ( ph  ->  S  e.  Cat )
268, 18, 25fuccat 15819 . . . . . . 7  |-  ( ph  ->  Q  e.  Cat )
27 eqid 2420 . . . . . . 7  |-  ( Q  2ndF  O )  =  ( Q  2ndF  O )
287, 26, 18, 272ndfcl 16027 . . . . . 6  |-  ( ph  ->  ( Q  2ndF  O )  e.  ( ( Q  X.c  O
)  Func  O )
)
29 eqid 2420 . . . . . . . 8  |-  (oppCat `  Q )  =  (oppCat `  Q )
30 relfunc 15711 . . . . . . . . 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 16091 . . . . . . . . 9  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
34 1st2ndbr 6847 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  ( 1st `  Y ) ( C  Func  Q )
( 2nd `  Y
) )
3530, 33, 34sylancr 667 . . . . . . . 8  |-  ( ph  ->  ( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
3610, 29, 35funcoppc 15724 . . . . . . 7  |-  ( ph  ->  ( 1st `  Y
) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y ) )
37 df-br 4418 . . . . . . 7  |-  ( ( 1st `  Y ) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y )  <->  <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  e.  ( O  Func  (oppCat `  Q ) ) )
3836, 37sylib 199 . . . . . 6  |-  ( ph  -> 
<. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  e.  ( O  Func  (oppCat `  Q
) ) )
3928, 38cofucl 15737 . . . . 5  |-  ( ph  ->  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) )  e.  ( ( Q  X.c  O ) 
Func  (oppCat `  Q )
) )
40 eqid 2420 . . . . . 6  |-  ( Q  1stF  O )  =  ( Q  1stF  O )
417, 26, 18, 401stfcl 16026 . . . . 5  |-  ( ph  ->  ( Q  1stF  O )  e.  ( ( Q  X.c  O
)  Func  Q )
)
4214, 15, 39, 41prfcl 16032 . . . 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 3640 . . . . 5  |-  ( ph  ->  ran  ( Hom f  `  Q ) 
C_  V )
4643, 29, 44, 26, 19, 45hofcl 16088 . . . 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 4877 . . . . 5  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  <. F ,  X >.  e.  ( ( O  Func  S )  X.  B ) )
5047, 48, 49syl2anc 665 . . . 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 4877 . . . . 5  |-  ( ( G  e.  ( O 
Func  S )  /\  P  e.  B )  ->  <. G ,  P >.  e.  ( ( O  Func  S )  X.  B ) )
5451, 52, 53syl2anc 665 . . . 4  |-  ( ph  -> 
<. G ,  P >.  e.  ( ( O  Func  S )  X.  B ) )
55 eqid 2420 . . . 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 2420 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
5958, 10oppchom 15564 . . . . . . 7  |-  ( X ( Hom  `  O
) P )  =  ( P ( Hom  `  C ) X )
6057, 59syl6eleqr 2519 . . . . . 6  |-  ( ph  ->  K  e.  ( X ( Hom  `  O
) P ) )
61 opelxpi 4877 . . . . . 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 665 . . . . 5  |-  ( ph  -> 
<. A ,  K >.  e.  ( ( F ( O Nat  S ) G )  X.  ( X ( Hom  `  O
) P ) ) )
63 eqid 2420 . . . . . . 7  |-  ( O Nat 
S )  =  ( O Nat  S )
648, 63fuchom 15810 . . . . . 6  |-  ( O Nat 
S )  =  ( Hom  `  Q )
65 eqid 2420 . . . . . 6  |-  ( Hom  `  O )  =  ( Hom  `  O )
667, 9, 12, 64, 65, 47, 48, 51, 52, 55xpchom2 16015 . . . . 5  |-  ( ph  ->  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. )  =  ( ( F ( O Nat  S ) G )  X.  ( X ( Hom  `  O
) P ) ) )
6762, 66eleqtrrd 2511 . . . 4  |-  ( ph  -> 
<. A ,  K >.  e.  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) )
6813, 42, 46, 50, 54, 55, 67cofu2 15735 . . 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 2473 . 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 16029 . . . . . 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 15733 . . . . . . . 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 5882 . . . . . . . . . . 11  |-  ( 1st `  Y )  e.  _V
73 fvex 5882 . . . . . . . . . . . 12  |-  ( 2nd `  Y )  e.  _V
7473tposex 7006 . . . . . . . . . . 11  |- tpos  ( 2nd `  Y )  e.  _V
7572, 74op1st 6806 . . . . . . . . . 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 16024 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  ( 2nd `  <. F ,  X >. )
)
78 op2ndg 6811 . . . . . . . . . . 11  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  ( 2nd `  <. F ,  X >. )  =  X )
7947, 48, 78syl2anc 665 . . . . . . . . . 10  |-  ( ph  ->  ( 2nd `  <. F ,  X >. )  =  X )
8077, 79eqtrd 2461 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  X )
8176, 80fveq12d 5878 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. ) `  (
( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) )  =  ( ( 1st `  Y ) `
 X ) )
8271, 81eqtrd 2461 . . . . . . 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 16021 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  ( 1st `  <. F ,  X >. )
)
84 op1stg 6810 . . . . . . . . 9  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  ( 1st `  <. F ,  X >. )  =  F )
8547, 48, 84syl2anc 665 . . . . . . . 8  |-  ( ph  ->  ( 1st `  <. F ,  X >. )  =  F )
8683, 85eqtrd 2461 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  F )
8782, 86opeq12d 4189 . . . . . 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 2461 . . . . 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 16029 . . . . . 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 15733 . . . . . . . 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 16024 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. )  =  ( 2nd `  <. G ,  P >. )
)
92 op2ndg 6811 . . . . . . . . . . 11  |-  ( ( G  e.  ( O 
Func  S )  /\  P  e.  B )  ->  ( 2nd `  <. G ,  P >. )  =  P )
9351, 52, 92syl2anc 665 . . . . . . . . . 10  |-  ( ph  ->  ( 2nd `  <. G ,  P >. )  =  P )
9491, 93eqtrd 2461 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. )  =  P )
9576, 94fveq12d 5878 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. ) `  (
( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. ) )  =  ( ( 1st `  Y ) `
 P ) )
9690, 95eqtrd 2461 . . . . . . 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 16021 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. G ,  P >. )  =  ( 1st `  <. G ,  P >. )
)
98 op1stg 6810 . . . . . . . . 9  |-  ( ( G  e.  ( O 
Func  S )  /\  P  e.  B )  ->  ( 1st `  <. G ,  P >. )  =  G )
9951, 52, 98syl2anc 665 . . . . . . . 8  |-  ( ph  ->  ( 1st `  <. G ,  P >. )  =  G )
10097, 99eqtrd 2461 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. G ,  P >. )  =  G )
10196, 100opeq12d 4189 . . . . . 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 2461 . . . . 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 6314 . . . 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 16031 . . . . 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 15735 . . . . . . 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 6807 . . . . . . . . . . 11  |-  ( 2nd `  <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >. )  = tpos  ( 2nd `  Y
)
107106oveqi 6309 . . . . . . . . . 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 6987 . . . . . . . . . 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 2449 . . . . . . . . 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 6314 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 1st `  ( Q  2ndF  O )
) `  <. G ,  P >. ) ( 2nd `  Y ) ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) )  =  ( P ( 2nd `  Y
) X ) )
111109, 110syl5eq 2473 . . . . . . . 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 16025 . . . . . . . . . 10  |-  ( ph  ->  ( <. F ,  X >. ( 2nd `  ( Q  2ndF  O ) ) <. G ,  P >. )  =  ( 2nd  |`  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) ) )
113112fveq1d 5874 . . . . . . . . 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 5886 . . . . . . . . . 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 17 . . . . . . . . 9  |-  ( ph  ->  ( ( 2nd  |`  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) ) `  <. A ,  K >. )  =  ( 2nd `  <. A ,  K >. ) )
116 op2ndg 6811 . . . . . . . . . 10  |-  ( ( A  e.  ( F ( O Nat  S ) G )  /\  K  e.  ( P ( Hom  `  C ) X ) )  ->  ( 2nd ` 
<. A ,  K >. )  =  K )
11756, 57, 116syl2anc 665 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  <. A ,  K >. )  =  K )
118113, 115, 1173eqtrd 2465 . . . . . . . 8  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  ( Q  2ndF  O ) ) <. G ,  P >. ) `
 <. A ,  K >. )  =  K )
119111, 118fveq12d 5878 . . . . . . 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 2461 . . . . . 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 16022 . . . . . . . 8  |-  ( ph  ->  ( <. F ,  X >. ( 2nd `  ( Q  1stF  O ) ) <. G ,  P >. )  =  ( 1st  |`  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) ) )
122121fveq1d 5874 . . . . . . 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 5886 . . . . . . . 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 17 . . . . . . 7  |-  ( ph  ->  ( ( 1st  |`  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) ) `  <. A ,  K >. )  =  ( 1st `  <. A ,  K >. ) )
125 op1stg 6810 . . . . . . . 8  |-  ( ( A  e.  ( F ( O Nat  S ) G )  /\  K  e.  ( P ( Hom  `  C ) X ) )  ->  ( 1st ` 
<. A ,  K >. )  =  A )
12656, 57, 125syl2anc 665 . . . . . . 7  |-  ( ph  ->  ( 1st `  <. A ,  K >. )  =  A )
127122, 124, 1263eqtrd 2465 . . . . . 6  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  ( Q  1stF  O ) ) <. G ,  P >. ) `
 <. A ,  K >. )  =  A )
128120, 127opeq12d 4189 . . . . 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 2461 . . . 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 5878 . . 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 6299 . . 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 2479 . 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 2461 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 1437    e. wcel 1867   _Vcvv 3078    u. cun 3431    C_ wss 3433   <.cop 3999   class class class wbr 4417    X. cxp 4843   ran crn 4846    |` cres 4847   Rel wrel 4850   ` cfv 5592  (class class class)co 6296   1stc1st 6796   2ndc2nd 6797  tpos ctpos 6971   Basecbs 15073   Hom chom 15153   Catccat 15514   Idccid 15515   Hom f chomf 15516  oppCatcoppc 15560    Func cfunc 15703    o.func ccofu 15705   Nat cnat 15790   FuncCat cfuc 15791   SetCatcsetc 15914    X.c cxpc 15997    1stF c1stf 15998    2ndF c2ndf 15999   ⟨,⟩F cprf 16000   evalF cevlf 16038  HomFchof 16077  Yoncyon 16078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-tpos 6972  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-fz 11772  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-hom 15166  df-cco 15167  df-cat 15518  df-cid 15519  df-homf 15520  df-comf 15521  df-oppc 15561  df-func 15707  df-cofu 15709  df-nat 15792  df-fuc 15793  df-setc 15915  df-xpc 16001  df-1stf 16002  df-2ndf 16003  df-prf 16004  df-curf 16043  df-hof 16079  df-yon 16080
This theorem is referenced by:  yonedalem3b  16108
  Copyright terms: Public domain W3C validator