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Theorem yonedalem22 14330
Description: Lemma for yoneda 14335. (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  (  Homf  `  C ) 
C_  U )
yoneda.v  |-  ( ph  ->  ( ran  (  Homf  `  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 5690 . . . . . 6  |-  ( 2nd `  Z )  =  ( 2nd `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) )
32oveqi 6053 . . . . 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 6053 . . . 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 6043 . . . 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 2424 . . 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 2404 . . . . 5  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
8 yoneda.q . . . . . 6  |-  Q  =  ( O FuncCat  S )
98fucbas 14112 . . . . 5  |-  ( O 
Func  S )  =  (
Base `  Q )
10 yoneda.o . . . . . 6  |-  O  =  (oppCat `  C )
11 yoneda.b . . . . . 6  |-  B  =  ( Base `  C
)
1210, 11oppcbas 13899 . . . . 5  |-  B  =  ( Base `  O
)
137, 9, 12xpcbas 14230 . . . 4  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
14 eqid 2404 . . . . 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 2404 . . . . 5  |-  ( (oppCat `  Q )  X.c  Q )  =  ( (oppCat `  Q )  X.c  Q )
16 yoneda.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
1710oppccat 13903 . . . . . . . . 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  (  Homf  `  Q )  u.  U
)  C_  V )
2120unssbd 3485 . . . . . . . . . 10  |-  ( ph  ->  U  C_  V )
2219, 21ssexd 4310 . . . . . . . . 9  |-  ( ph  ->  U  e.  _V )
23 yoneda.s . . . . . . . . . 10  |-  S  =  ( SetCat `  U )
2423setccat 14195 . . . . . . . . 9  |-  ( U  e.  _V  ->  S  e.  Cat )
2522, 24syl 16 . . . . . . . 8  |-  ( ph  ->  S  e.  Cat )
268, 18, 25fuccat 14122 . . . . . . 7  |-  ( ph  ->  Q  e.  Cat )
27 eqid 2404 . . . . . . 7  |-  ( Q  2ndF  O )  =  ( Q  2ndF  O )
287, 26, 18, 272ndfcl 14250 . . . . . 6  |-  ( ph  ->  ( Q  2ndF  O )  e.  ( ( Q  X.c  O
)  Func  O )
)
29 eqid 2404 . . . . . . . 8  |-  (oppCat `  Q )  =  (oppCat `  Q )
30 relfunc 14014 . . . . . . . . 9  |-  Rel  ( C  Func  Q )
31 yoneda.y . . . . . . . . . 10  |-  Y  =  (Yon `  C )
32 yoneda.u . . . . . . . . . 10  |-  ( ph  ->  ran  (  Homf  `  C ) 
C_  U )
3331, 16, 10, 23, 8, 22, 32yoncl 14314 . . . . . . . . 9  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
34 1st2ndbr 6355 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  ( 1st `  Y ) ( C  Func  Q )
( 2nd `  Y
) )
3530, 33, 34sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
3610, 29, 35funcoppc 14027 . . . . . . 7  |-  ( ph  ->  ( 1st `  Y
) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y ) )
37 df-br 4173 . . . . . . 7  |-  ( ( 1st `  Y ) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y )  <->  <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  e.  ( O  Func  (oppCat `  Q ) ) )
3836, 37sylib 189 . . . . . 6  |-  ( ph  -> 
<. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  e.  ( O  Func  (oppCat `  Q
) ) )
3928, 38cofucl 14040 . . . . 5  |-  ( ph  ->  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) )  e.  ( ( Q  X.c  O ) 
Func  (oppCat `  Q )
) )
40 eqid 2404 . . . . . 6  |-  ( Q  1stF  O )  =  ( Q  1stF  O )
417, 26, 18, 401stfcl 14249 . . . . 5  |-  ( ph  ->  ( Q  1stF  O )  e.  ( ( Q  X.c  O
)  Func  Q )
)
4214, 15, 39, 41prfcl 14255 . . . 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 3484 . . . . 5  |-  ( ph  ->  ran  (  Homf  `  Q ) 
C_  V )
4643, 29, 44, 26, 19, 45hofcl 14311 . . . 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 4869 . . . . 5  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  <. F ,  X >.  e.  ( ( O  Func  S )  X.  B ) )
5047, 48, 49syl2anc 643 . . . 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 4869 . . . . 5  |-  ( ( G  e.  ( O 
Func  S )  /\  P  e.  B )  ->  <. G ,  P >.  e.  ( ( O  Func  S )  X.  B ) )
5451, 52, 53syl2anc 643 . . . 4  |-  ( ph  -> 
<. G ,  P >.  e.  ( ( O  Func  S )  X.  B ) )
55 eqid 2404 . . . 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 2404 . . . . . . . 8  |-  (  Hom  `  C )  =  (  Hom  `  C )
5958, 10oppchom 13896 . . . . . . 7  |-  ( X (  Hom  `  O
) P )  =  ( P (  Hom  `  C ) X )
6057, 59syl6eleqr 2495 . . . . . 6  |-  ( ph  ->  K  e.  ( X (  Hom  `  O
) P ) )
61 opelxpi 4869 . . . . . 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 643 . . . . 5  |-  ( ph  -> 
<. A ,  K >.  e.  ( ( F ( O Nat  S ) G )  X.  ( X (  Hom  `  O
) P ) ) )
63 eqid 2404 . . . . . . 7  |-  ( O Nat 
S )  =  ( O Nat  S )
648, 63fuchom 14113 . . . . . 6  |-  ( O Nat 
S )  =  (  Hom  `  Q )
65 eqid 2404 . . . . . 6  |-  (  Hom  `  O )  =  (  Hom  `  O )
667, 9, 12, 64, 65, 47, 48, 51, 52, 55xpchom2 14238 . . . . 5  |-  ( ph  ->  ( <. F ,  X >. (  Hom  `  ( Q  X.c  O ) ) <. G ,  P >. )  =  ( ( F ( O Nat  S ) G )  X.  ( X (  Hom  `  O
) P ) ) )
6762, 66eleqtrrd 2481 . . . 4  |-  ( ph  -> 
<. A ,  K >.  e.  ( <. F ,  X >. (  Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) )
6813, 42, 46, 50, 54, 55, 67cofu2 14038 . . 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 2448 . 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 14252 . . . . . 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 14036 . . . . . . . 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 5701 . . . . . . . . . . 11  |-  ( 1st `  Y )  e.  _V
73 fvex 5701 . . . . . . . . . . . 12  |-  ( 2nd `  Y )  e.  _V
7473tposex 6472 . . . . . . . . . . 11  |- tpos  ( 2nd `  Y )  e.  _V
7572, 74op1st 6314 . . . . . . . . . 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 14247 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  ( 2nd `  <. F ,  X >. )
)
78 op2ndg 6319 . . . . . . . . . . 11  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  ( 2nd `  <. F ,  X >. )  =  X )
7947, 48, 78syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( 2nd `  <. F ,  X >. )  =  X )
8077, 79eqtrd 2436 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  X )
8176, 80fveq12d 5693 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. ) `  (
( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) )  =  ( ( 1st `  Y ) `
 X ) )
8271, 81eqtrd 2436 . . . . . . 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 14244 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  ( 1st `  <. F ,  X >. )
)
84 op1stg 6318 . . . . . . . . 9  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  ( 1st `  <. F ,  X >. )  =  F )
8547, 48, 84syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( 1st `  <. F ,  X >. )  =  F )
8683, 85eqtrd 2436 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  F )
8782, 86opeq12d 3952 . . . . . 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 2436 . . . . 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 14252 . . . . . 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 14036 . . . . . . . 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 14247 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. )  =  ( 2nd `  <. G ,  P >. )
)
92 op2ndg 6319 . . . . . . . . . . 11  |-  ( ( G  e.  ( O 
Func  S )  /\  P  e.  B )  ->  ( 2nd `  <. G ,  P >. )  =  P )
9351, 52, 92syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( 2nd `  <. G ,  P >. )  =  P )
9491, 93eqtrd 2436 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. )  =  P )
9576, 94fveq12d 5693 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. ) `  (
( 1st `  ( Q  2ndF  O ) ) `  <. G ,  P >. ) )  =  ( ( 1st `  Y ) `
 P ) )
9690, 95eqtrd 2436 . . . . . . 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 14244 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. G ,  P >. )  =  ( 1st `  <. G ,  P >. )
)
98 op1stg 6318 . . . . . . . . 9  |-  ( ( G  e.  ( O 
Func  S )  /\  P  e.  B )  ->  ( 1st `  <. G ,  P >. )  =  G )
9951, 52, 98syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( 1st `  <. G ,  P >. )  =  G )
10097, 99eqtrd 2436 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. G ,  P >. )  =  G )
10196, 100opeq12d 3952 . . . . . 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 2436 . . . . 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 6058 . . . 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 14254 . . . . 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 14038 . . . . . . 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 6315 . . . . . . . . . . 11  |-  ( 2nd `  <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >. )  = tpos  ( 2nd `  Y
)
107106oveqi 6053 . . . . . . . . . 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 6453 . . . . . . . . . 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 2424 . . . . . . . . 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 6058 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 1st `  ( Q  2ndF  O )
) `  <. G ,  P >. ) ( 2nd `  Y ) ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) )  =  ( P ( 2nd `  Y
) X ) )
111109, 110syl5eq 2448 . . . . . . . 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 14248 . . . . . . . . . 10  |-  ( ph  ->  ( <. F ,  X >. ( 2nd `  ( Q  2ndF  O ) ) <. G ,  P >. )  =  ( 2nd  |`  ( <. F ,  X >. (  Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) ) )
113112fveq1d 5689 . . . . . . . . 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 5704 . . . . . . . . . 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 6319 . . . . . . . . . 10  |-  ( ( A  e.  ( F ( O Nat  S ) G )  /\  K  e.  ( P (  Hom  `  C ) X ) )  ->  ( 2nd ` 
<. A ,  K >. )  =  K )
11756, 57, 116syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  <. A ,  K >. )  =  K )
118113, 115, 1173eqtrd 2440 . . . . . . . 8  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  ( Q  2ndF  O ) ) <. G ,  P >. ) `
 <. A ,  K >. )  =  K )
119111, 118fveq12d 5693 . . . . . . 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 2436 . . . . . 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 14245 . . . . . . . 8  |-  ( ph  ->  ( <. F ,  X >. ( 2nd `  ( Q  1stF  O ) ) <. G ,  P >. )  =  ( 1st  |`  ( <. F ,  X >. (  Hom  `  ( Q  X.c  O ) ) <. G ,  P >. ) ) )
122121fveq1d 5689 . . . . . . 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 5704 . . . . . . . 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 6318 . . . . . . . 8  |-  ( ( A  e.  ( F ( O Nat  S ) G )  /\  K  e.  ( P (  Hom  `  C ) X ) )  ->  ( 1st ` 
<. A ,  K >. )  =  A )
12656, 57, 125syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( 1st `  <. A ,  K >. )  =  A )
127122, 124, 1263eqtrd 2440 . . . . . 6  |-  ( ph  ->  ( ( <. F ,  X >. ( 2nd `  ( Q  1stF  O ) ) <. G ,  P >. ) `
 <. A ,  K >. )  =  A )
128120, 127opeq12d 3952 . . . . 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 2436 . . . 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 5693 . . 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 6043 . . 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 2454 . 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 2436 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 set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278    C_ wss 3280   <.cop 3777   class class class wbr 4172    X. cxp 4835   ran crn 4838    |` cres 4839   Rel wrel 4842   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307  tpos ctpos 6437   Basecbs 13424    Hom chom 13495   Catccat 13844   Idccid 13845    Homf chomf 13846  oppCatcoppc 13892    Func cfunc 14006    o.func ccofu 14008   Nat cnat 14093   FuncCat cfuc 14094   SetCatcsetc 14185    X.c cxpc 14220    1stF c1stf 14221    2ndF c2ndf 14222   ⟨,⟩F cprf 14223   evalF cevlf 14261  HomFchof 14300  Yoncyon 14301
This theorem is referenced by:  yonedalem3b  14331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-hom 13508  df-cco 13509  df-cat 13848  df-cid 13849  df-homf 13850  df-comf 13851  df-oppc 13893  df-func 14010  df-cofu 14012  df-nat 14095  df-fuc 14096  df-setc 14186  df-xpc 14224  df-1stf 14225  df-2ndf 14226  df-prf 14227  df-curf 14266  df-hof 14302  df-yon 14303
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