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Theorem yonedalem22 15673
Description: Lemma for yoneda 15678. (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 2486 . . 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 2457 . . . . 5  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
8 yoneda.q . . . . . 6  |-  Q  =  ( O FuncCat  S )
98fucbas 15375 . . . . 5  |-  ( O 
Func  S )  =  (
Base `  Q )
10 yoneda.o . . . . . 6  |-  O  =  (oppCat `  C )
11 yoneda.b . . . . . 6  |-  B  =  ( Base `  C
)
1210, 11oppcbas 15133 . . . . 5  |-  B  =  ( Base `  O
)
137, 9, 12xpcbas 15573 . . . 4  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
14 eqid 2457 . . . . 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 2457 . . . . 5  |-  ( (oppCat `  Q )  X.c  Q )  =  ( (oppCat `  Q )  X.c  Q )
16 yoneda.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
1710oppccat 15137 . . . . . . . . 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 3678 . . . . . . . . . 10  |-  ( ph  ->  U  C_  V )
2219, 21ssexd 4603 . . . . . . . . 9  |-  ( ph  ->  U  e.  _V )
23 yoneda.s . . . . . . . . . 10  |-  S  =  ( SetCat `  U )
2423setccat 15490 . . . . . . . . 9  |-  ( U  e.  _V  ->  S  e.  Cat )
2522, 24syl 16 . . . . . . . 8  |-  ( ph  ->  S  e.  Cat )
268, 18, 25fuccat 15385 . . . . . . 7  |-  ( ph  ->  Q  e.  Cat )
27 eqid 2457 . . . . . . 7  |-  ( Q  2ndF  O )  =  ( Q  2ndF  O )
287, 26, 18, 272ndfcl 15593 . . . . . 6  |-  ( ph  ->  ( Q  2ndF  O )  e.  ( ( Q  X.c  O
)  Func  O )
)
29 eqid 2457 . . . . . . . 8  |-  (oppCat `  Q )  =  (oppCat `  Q )
30 relfunc 15277 . . . . . . . . 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 15657 . . . . . . . . 9  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
34 1st2ndbr 6848 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  ( 1st `  Y ) ( C  Func  Q )
( 2nd `  Y
) )
3530, 33, 34sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
3610, 29, 35funcoppc 15290 . . . . . . 7  |-  ( ph  ->  ( 1st `  Y
) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y ) )
37 df-br 4457 . . . . . . 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 15303 . . . . 5  |-  ( ph  ->  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) )  e.  ( ( Q  X.c  O ) 
Func  (oppCat `  Q )
) )
40 eqid 2457 . . . . . 6  |-  ( Q  1stF  O )  =  ( Q  1stF  O )
417, 26, 18, 401stfcl 15592 . . . . 5  |-  ( ph  ->  ( Q  1stF  O )  e.  ( ( Q  X.c  O
)  Func  Q )
)
4214, 15, 39, 41prfcl 15598 . . . 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 3677 . . . . 5  |-  ( ph  ->  ran  ( Hom f  `  Q ) 
C_  V )
4643, 29, 44, 26, 19, 45hofcl 15654 . . . 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 5040 . . . . 5  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  <. F ,  X >.  e.  ( ( O  Func  S )  X.  B ) )
5047, 48, 49syl2anc 661 . . . 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 5040 . . . . 5  |-  ( ( G  e.  ( O 
Func  S )  /\  P  e.  B )  ->  <. G ,  P >.  e.  ( ( O  Func  S )  X.  B ) )
5451, 52, 53syl2anc 661 . . . 4  |-  ( ph  -> 
<. G ,  P >.  e.  ( ( O  Func  S )  X.  B ) )
55 eqid 2457 . . . 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 2457 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
5958, 10oppchom 15130 . . . . . . 7  |-  ( X ( Hom  `  O
) P )  =  ( P ( Hom  `  C ) X )
6057, 59syl6eleqr 2556 . . . . . 6  |-  ( ph  ->  K  e.  ( X ( Hom  `  O
) P ) )
61 opelxpi 5040 . . . . . 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 661 . . . . 5  |-  ( ph  -> 
<. A ,  K >.  e.  ( ( F ( O Nat  S ) G )  X.  ( X ( Hom  `  O
) P ) ) )
63 eqid 2457 . . . . . . 7  |-  ( O Nat 
S )  =  ( O Nat  S )
648, 63fuchom 15376 . . . . . 6  |-  ( O Nat 
S )  =  ( Hom  `  Q )
65 eqid 2457 . . . . . 6  |-  ( Hom  `  O )  =  ( Hom  `  O )
667, 9, 12, 64, 65, 47, 48, 51, 52, 55xpchom2 15581 . . . . 5  |-  ( ph  ->  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. )  =  ( ( F ( O Nat  S ) G )  X.  ( X ( Hom  `  O
) P ) ) )
6762, 66eleqtrrd 2548 . . . 4  |-  ( ph  -> 
<. A ,  K >.  e.  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) )
6813, 42, 46, 50, 54, 55, 67cofu2 15301 . . 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 2510 . 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 15595 . . . . . 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 15299 . . . . . . . 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 7007 . . . . . . . . . . 11  |- tpos  ( 2nd `  Y )  e.  _V
7572, 74op1st 6807 . . . . . . . . . 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 15590 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  ( 2nd `  <. F ,  X >. )
)
78 op2ndg 6812 . . . . . . . . . . 11  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  ( 2nd `  <. F ,  X >. )  =  X )
7947, 48, 78syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( 2nd `  <. F ,  X >. )  =  X )
8077, 79eqtrd 2498 . . . . . . . . 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 2498 . . . . . . 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 15587 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  ( 1st `  <. F ,  X >. )
)
84 op1stg 6811 . . . . . . . . 9  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  ( 1st `  <. F ,  X >. )  =  F )
8547, 48, 84syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( 1st `  <. F ,  X >. )  =  F )
8683, 85eqtrd 2498 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  F )
8782, 86opeq12d 4227 . . . . . 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 2498 . . . . 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 15595 . . . . . 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 15299 . . . . . . . 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 15590 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. )  =  ( 2nd `  <. G ,  P >. )
)
92 op2ndg 6812 . . . . . . . . . . 11  |-  ( ( G  e.  ( O 
Func  S )  /\  P  e.  B )  ->  ( 2nd `  <. G ,  P >. )  =  P )
9351, 52, 92syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( 2nd `  <. G ,  P >. )  =  P )
9491, 93eqtrd 2498 . . . . . . . . 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 2498 . . . . . . 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 15587 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. G ,  P >. )  =  ( 1st `  <. G ,  P >. )
)
98 op1stg 6811 . . . . . . . . 9  |-  ( ( G  e.  ( O 
Func  S )  /\  P  e.  B )  ->  ( 1st `  <. G ,  P >. )  =  G )
9951, 52, 98syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( 1st `  <. G ,  P >. )  =  G )
10097, 99eqtrd 2498 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. G ,  P >. )  =  G )
10196, 100opeq12d 4227 . . . . . 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 2498 . . . . 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 15597 . . . . 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 15301 . . . . . . 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 6808 . . . . . . . . . . 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 6988 . . . . . . . . . 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 2486 . . . . . . . . 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 2510 . . . . . . . 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 15591 . . . . . . . . . 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 16 . . . . . . . . 9  |-  ( ph  ->  ( ( 2nd  |`  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) ) `  <. A ,  K >. )  =  ( 2nd `  <. A ,  K >. ) )
116 op2ndg 6812 . . . . . . . . . 10  |-  ( ( A  e.  ( F ( O Nat  S ) G )  /\  K  e.  ( P ( Hom  `  C ) X ) )  ->  ( 2nd ` 
<. A ,  K >. )  =  K )
11756, 57, 116syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  <. A ,  K >. )  =  K )
118113, 115, 1173eqtrd 2502 . . . . . . . 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 2498 . . . . . 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 15588 . . . . . . . 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 16 . . . . . . 7  |-  ( ph  ->  ( ( 1st  |`  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) ) `  <. A ,  K >. )  =  ( 1st `  <. A ,  K >. ) )
125 op1stg 6811 . . . . . . . 8  |-  ( ( A  e.  ( F ( O Nat  S ) G )  /\  K  e.  ( P ( Hom  `  C ) X ) )  ->  ( 1st ` 
<. A ,  K >. )  =  A )
12656, 57, 125syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( 1st `  <. A ,  K >. )  =  A )
127122, 124, 1263eqtrd 2502 . . . . . 6  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  ( Q  1stF  O ) ) <. G ,  P >. ) `
 <. A ,  K >. )  =  A )
128120, 127opeq12d 4227 . . . . 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 2498 . . . 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 2516 . 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 2498 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 1395    e. wcel 1819   _Vcvv 3109    u. cun 3469    C_ wss 3471   <.cop 4038   class class class wbr 4456    X. cxp 5006   ran crn 5009    |` cres 5010   Rel wrel 5013   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798  tpos ctpos 6972   Basecbs 14643   Hom chom 14722   Catccat 15080   Idccid 15081   Hom f chomf 15082  oppCatcoppc 15126    Func cfunc 15269    o.func ccofu 15271   Nat cnat 15356   FuncCat cfuc 15357   SetCatcsetc 15480    X.c cxpc 15563    1stF c1stf 15564    2ndF c2ndf 15565   ⟨,⟩F cprf 15566   evalF cevlf 15604  HomFchof 15643  Yoncyon 15644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-hom 14735  df-cco 14736  df-cat 15084  df-cid 15085  df-homf 15086  df-comf 15087  df-oppc 15127  df-func 15273  df-cofu 15275  df-nat 15358  df-fuc 15359  df-setc 15481  df-xpc 15567  df-1stf 15568  df-2ndf 15569  df-prf 15570  df-curf 15609  df-hof 15645  df-yon 15646
This theorem is referenced by:  yonedalem3b  15674
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