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Theorem yonedalem22 15103
Description: Lemma for yoneda 15108. (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 5709 . . . . . 6  |-  ( 2nd `  Z )  =  ( 2nd `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) )
32oveqi 6119 . . . . 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 6119 . . . 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 6109 . . . 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 2463 . . 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 2443 . . . . 5  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
8 yoneda.q . . . . . 6  |-  Q  =  ( O FuncCat  S )
98fucbas 14885 . . . . 5  |-  ( O 
Func  S )  =  (
Base `  Q )
10 yoneda.o . . . . . 6  |-  O  =  (oppCat `  C )
11 yoneda.b . . . . . 6  |-  B  =  ( Base `  C
)
1210, 11oppcbas 14672 . . . . 5  |-  B  =  ( Base `  O
)
137, 9, 12xpcbas 15003 . . . 4  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
14 eqid 2443 . . . . 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 2443 . . . . 5  |-  ( (oppCat `  Q )  X.c  Q )  =  ( (oppCat `  Q )  X.c  Q )
16 yoneda.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
1710oppccat 14676 . . . . . . . . 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 3549 . . . . . . . . . 10  |-  ( ph  ->  U  C_  V )
2219, 21ssexd 4454 . . . . . . . . 9  |-  ( ph  ->  U  e.  _V )
23 yoneda.s . . . . . . . . . 10  |-  S  =  ( SetCat `  U )
2423setccat 14968 . . . . . . . . 9  |-  ( U  e.  _V  ->  S  e.  Cat )
2522, 24syl 16 . . . . . . . 8  |-  ( ph  ->  S  e.  Cat )
268, 18, 25fuccat 14895 . . . . . . 7  |-  ( ph  ->  Q  e.  Cat )
27 eqid 2443 . . . . . . 7  |-  ( Q  2ndF  O )  =  ( Q  2ndF  O )
287, 26, 18, 272ndfcl 15023 . . . . . 6  |-  ( ph  ->  ( Q  2ndF  O )  e.  ( ( Q  X.c  O
)  Func  O )
)
29 eqid 2443 . . . . . . . 8  |-  (oppCat `  Q )  =  (oppCat `  Q )
30 relfunc 14787 . . . . . . . . 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 15087 . . . . . . . . 9  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
34 1st2ndbr 6638 . . . . . . . . 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 14800 . . . . . . 7  |-  ( ph  ->  ( 1st `  Y
) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y ) )
37 df-br 4308 . . . . . . 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 14813 . . . . 5  |-  ( ph  ->  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) )  e.  ( ( Q  X.c  O ) 
Func  (oppCat `  Q )
) )
40 eqid 2443 . . . . . 6  |-  ( Q  1stF  O )  =  ( Q  1stF  O )
417, 26, 18, 401stfcl 15022 . . . . 5  |-  ( ph  ->  ( Q  1stF  O )  e.  ( ( Q  X.c  O
)  Func  Q )
)
4214, 15, 39, 41prfcl 15028 . . . 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 3548 . . . . 5  |-  ( ph  ->  ran  ( Hom f  `  Q ) 
C_  V )
4643, 29, 44, 26, 19, 45hofcl 15084 . . . 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 4886 . . . . 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 4886 . . . . 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 2443 . . . 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 2443 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
5958, 10oppchom 14669 . . . . . . 7  |-  ( X ( Hom  `  O
) P )  =  ( P ( Hom  `  C ) X )
6057, 59syl6eleqr 2534 . . . . . 6  |-  ( ph  ->  K  e.  ( X ( Hom  `  O
) P ) )
61 opelxpi 4886 . . . . . 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 2443 . . . . . . 7  |-  ( O Nat 
S )  =  ( O Nat  S )
648, 63fuchom 14886 . . . . . 6  |-  ( O Nat 
S )  =  ( Hom  `  Q )
65 eqid 2443 . . . . . 6  |-  ( Hom  `  O )  =  ( Hom  `  O )
667, 9, 12, 64, 65, 47, 48, 51, 52, 55xpchom2 15011 . . . . 5  |-  ( ph  ->  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. )  =  ( ( F ( O Nat  S ) G )  X.  ( X ( Hom  `  O
) P ) ) )
6762, 66eleqtrrd 2520 . . . 4  |-  ( ph  -> 
<. A ,  K >.  e.  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) )
6813, 42, 46, 50, 54, 55, 67cofu2 14811 . . 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 2487 . 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 15025 . . . . . 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 14809 . . . . . . . 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 5716 . . . . . . . . . . 11  |-  ( 1st `  Y )  e.  _V
73 fvex 5716 . . . . . . . . . . . 12  |-  ( 2nd `  Y )  e.  _V
7473tposex 6794 . . . . . . . . . . 11  |- tpos  ( 2nd `  Y )  e.  _V
7572, 74op1st 6600 . . . . . . . . . 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 15020 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  ( 2nd `  <. F ,  X >. )
)
78 op2ndg 6605 . . . . . . . . . . 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 2475 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  X )
8176, 80fveq12d 5712 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. ) `  (
( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) )  =  ( ( 1st `  Y ) `
 X ) )
8271, 81eqtrd 2475 . . . . . . 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 15017 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  ( 1st `  <. F ,  X >. )
)
84 op1stg 6604 . . . . . . . . 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 2475 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  F )
8782, 86opeq12d 4082 . . . . . 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 2475 . . . . 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 15025 . . . . . 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 14809 . . . . . . . 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 15020 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. )  =  ( 2nd `  <. G ,  P >. )
)
92 op2ndg 6605 . . . . . . . . . . 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 2475 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. )  =  P )
9576, 94fveq12d 5712 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. ) `  (
( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. ) )  =  ( ( 1st `  Y ) `
 P ) )
9690, 95eqtrd 2475 . . . . . . 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 15017 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. G ,  P >. )  =  ( 1st `  <. G ,  P >. )
)
98 op1stg 6604 . . . . . . . . 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 2475 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. G ,  P >. )  =  G )
10196, 100opeq12d 4082 . . . . . 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 2475 . . . . 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 6124 . . . 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 15027 . . . . 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 14811 . . . . . . 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 6601 . . . . . . . . . . 11  |-  ( 2nd `  <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >. )  = tpos  ( 2nd `  Y
)
107106oveqi 6119 . . . . . . . . . 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 6775 . . . . . . . . . 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 2463 . . . . . . . . 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 6124 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 1st `  ( Q  2ndF  O )
) `  <. G ,  P >. ) ( 2nd `  Y ) ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) )  =  ( P ( 2nd `  Y
) X ) )
111109, 110syl5eq 2487 . . . . . . . 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 15021 . . . . . . . . . 10  |-  ( ph  ->  ( <. F ,  X >. ( 2nd `  ( Q  2ndF  O ) ) <. G ,  P >. )  =  ( 2nd  |`  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) ) )
113112fveq1d 5708 . . . . . . . . 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 5719 . . . . . . . . . 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 6605 . . . . . . . . . 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 2479 . . . . . . . 8  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  ( Q  2ndF  O ) ) <. G ,  P >. ) `
 <. A ,  K >. )  =  K )
119111, 118fveq12d 5712 . . . . . . 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 2475 . . . . . 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 15018 . . . . . . . 8  |-  ( ph  ->  ( <. F ,  X >. ( 2nd `  ( Q  1stF  O ) ) <. G ,  P >. )  =  ( 1st  |`  ( <. F ,  X >. ( Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) ) )
122121fveq1d 5708 . . . . . . 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 5719 . . . . . . . 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 6604 . . . . . . . 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 2479 . . . . . 6  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  ( Q  1stF  O ) ) <. G ,  P >. ) `
 <. A ,  K >. )  =  A )
128120, 127opeq12d 4082 . . . . 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 2475 . . . 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 5712 . . 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 6109 . . 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 2493 . 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 2475 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 1369    e. wcel 1756   _Vcvv 2987    u. cun 3341    C_ wss 3343   <.cop 3898   class class class wbr 4307    X. cxp 4853   ran crn 4856    |` cres 4857   Rel wrel 4860   ` cfv 5433  (class class class)co 6106   1stc1st 6590   2ndc2nd 6591  tpos ctpos 6759   Basecbs 14189   Hom chom 14264   Catccat 14617   Idccid 14618   Hom f chomf 14619  oppCatcoppc 14665    Func cfunc 14779    o.func ccofu 14781   Nat cnat 14866   FuncCat cfuc 14867   SetCatcsetc 14958    X.c cxpc 14993    1stF c1stf 14994    2ndF c2ndf 14995   ⟨,⟩F cprf 14996   evalF cevlf 15034  HomFchof 15073  Yoncyon 15074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-tpos 6760  df-recs 6847  df-rdg 6881  df-1o 6935  df-oadd 6939  df-er 7116  df-map 7231  df-ixp 7279  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-nn 10338  df-2 10395  df-3 10396  df-4 10397  df-5 10398  df-6 10399  df-7 10400  df-8 10401  df-9 10402  df-10 10403  df-n0 10595  df-z 10662  df-dec 10771  df-uz 10877  df-fz 11453  df-struct 14191  df-ndx 14192  df-slot 14193  df-base 14194  df-sets 14195  df-hom 14277  df-cco 14278  df-cat 14621  df-cid 14622  df-homf 14623  df-comf 14624  df-oppc 14666  df-func 14783  df-cofu 14785  df-nat 14868  df-fuc 14869  df-setc 14959  df-xpc 14997  df-1stf 14998  df-2ndf 14999  df-prf 15000  df-curf 15039  df-hof 15075  df-yon 15076
This theorem is referenced by:  yonedalem3b  15104
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