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Theorem yonedalem21 14049
Description: Lemma for yoneda 14059. (Contributed by Mario Carneiro, 28-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 )
Assertion
Ref Expression
yonedalem21  |-  ( ph  ->  ( F ( 1st `  Z ) X )  =  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F ) )

Proof of Theorem yonedalem21
StepHypRef Expression
1 yoneda.z . . . . . 6  |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )
21fveq2i 5530 . . . . 5  |-  ( 1st `  Z )  =  ( 1st `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) )
32oveqi 5873 . . . 4  |-  ( F ( 1st `  Z
) X )  =  ( F ( 1st `  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) ) X )
4 df-ov 5863 . . . 4  |-  ( F ( 1st `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) ) X )  =  ( ( 1st `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) ) `  <. F ,  X >. )
53, 4eqtri 2305 . . 3  |-  ( F ( 1st `  Z
) X )  =  ( ( 1st `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) ) `  <. F ,  X >. )
6 eqid 2285 . . . . 5  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
7 yoneda.q . . . . . 6  |-  Q  =  ( O FuncCat  S )
87fucbas 13836 . . . . 5  |-  ( O 
Func  S )  =  (
Base `  Q )
9 yoneda.o . . . . . 6  |-  O  =  (oppCat `  C )
10 yoneda.b . . . . . 6  |-  B  =  ( Base `  C
)
119, 10oppcbas 13623 . . . . 5  |-  B  =  ( Base `  O
)
126, 8, 11xpcbas 13954 . . . 4  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
13 eqid 2285 . . . . 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 )
)
14 eqid 2285 . . . . 5  |-  ( (oppCat `  Q )  X.c  Q )  =  ( (oppCat `  Q )  X.c  Q )
15 yoneda.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
169oppccat 13627 . . . . . . . . 9  |-  ( C  e.  Cat  ->  O  e.  Cat )
1715, 16syl 15 . . . . . . . 8  |-  ( ph  ->  O  e.  Cat )
18 ssun2 3341 . . . . . . . . . . 11  |-  U  C_  ( ran  (  Homf  `  Q )  u.  U )
19 yoneda.v . . . . . . . . . . 11  |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U
)  C_  V )
2018, 19syl5ss 3192 . . . . . . . . . 10  |-  ( ph  ->  U  C_  V )
21 yoneda.w . . . . . . . . . 10  |-  ( ph  ->  V  e.  W )
22 ssexg 4162 . . . . . . . . . 10  |-  ( ( U  C_  V  /\  V  e.  W )  ->  U  e.  _V )
2320, 21, 22syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  U  e.  _V )
24 yoneda.s . . . . . . . . . 10  |-  S  =  ( SetCat `  U )
2524setccat 13919 . . . . . . . . 9  |-  ( U  e.  _V  ->  S  e.  Cat )
2623, 25syl 15 . . . . . . . 8  |-  ( ph  ->  S  e.  Cat )
277, 17, 26fuccat 13846 . . . . . . 7  |-  ( ph  ->  Q  e.  Cat )
28 eqid 2285 . . . . . . 7  |-  ( Q  2ndF  O )  =  ( Q  2ndF  O )
296, 27, 17, 282ndfcl 13974 . . . . . 6  |-  ( ph  ->  ( Q  2ndF  O )  e.  ( ( Q  X.c  O
)  Func  O )
)
30 eqid 2285 . . . . . . . 8  |-  (oppCat `  Q )  =  (oppCat `  Q )
31 relfunc 13738 . . . . . . . . 9  |-  Rel  ( C  Func  Q )
32 yoneda.y . . . . . . . . . 10  |-  Y  =  (Yon `  C )
33 yoneda.u . . . . . . . . . 10  |-  ( ph  ->  ran  (  Homf  `  C ) 
C_  U )
3432, 15, 9, 24, 7, 23, 33yoncl 14038 . . . . . . . . 9  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
35 1st2ndbr 6171 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  ( 1st `  Y ) ( C  Func  Q )
( 2nd `  Y
) )
3631, 34, 35sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
379, 30, 36funcoppc 13751 . . . . . . 7  |-  ( ph  ->  ( 1st `  Y
) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y ) )
38 df-br 4026 . . . . . . 7  |-  ( ( 1st `  Y ) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y )  <->  <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  e.  ( O  Func  (oppCat `  Q ) ) )
3937, 38sylib 188 . . . . . 6  |-  ( ph  -> 
<. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  e.  ( O  Func  (oppCat `  Q
) ) )
4029, 39cofucl 13764 . . . . 5  |-  ( ph  ->  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) )  e.  ( ( Q  X.c  O ) 
Func  (oppCat `  Q )
) )
41 eqid 2285 . . . . . 6  |-  ( Q  1stF  O )  =  ( Q  1stF  O )
426, 27, 17, 411stfcl 13973 . . . . 5  |-  ( ph  ->  ( Q  1stF  O )  e.  ( ( Q  X.c  O
)  Func  Q )
)
4313, 14, 40, 42prfcl 13979 . . . 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 ) ) )
44 yoneda.h . . . . 5  |-  H  =  (HomF
`  Q )
45 yoneda.t . . . . 5  |-  T  =  ( SetCat `  V )
46 ssun1 3340 . . . . . 6  |-  ran  (  Homf  `  Q )  C_  ( ran  (  Homf 
`  Q )  u.  U )
4746, 19syl5ss 3192 . . . . 5  |-  ( ph  ->  ran  (  Homf  `  Q ) 
C_  V )
4844, 30, 45, 27, 21, 47hofcl 14035 . . . 4  |-  ( ph  ->  H  e.  ( ( (oppCat `  Q )  X.c  Q )  Func  T
) )
49 yonedalem21.f . . . . 5  |-  ( ph  ->  F  e.  ( O 
Func  S ) )
50 yonedalem21.x . . . . 5  |-  ( ph  ->  X  e.  B )
51 opelxpi 4723 . . . . 5  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  <. F ,  X >.  e.  ( ( O  Func  S )  X.  B ) )
5249, 50, 51syl2anc 642 . . . 4  |-  ( ph  -> 
<. F ,  X >.  e.  ( ( O  Func  S )  X.  B ) )
5312, 43, 48, 52cofu1 13760 . . 3  |-  ( ph  ->  ( ( 1st `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) ) `  <. F ,  X >. )  =  ( ( 1st `  H
) `  ( ( 1st `  ( ( <.
( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) `  <. F ,  X >. )
) )
545, 53syl5eq 2329 . 2  |-  ( ph  ->  ( F ( 1st `  Z ) X )  =  ( ( 1st `  H ) `  (
( 1st `  (
( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) `  <. F ,  X >. )
) )
55 eqid 2285 . . . . . 6  |-  (  Hom  `  ( Q  X.c  O ) )  =  (  Hom  `  ( Q  X.c  O ) )
5613, 12, 55, 40, 42, 52prf1 13976 . . . . 5  |-  ( 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 >. ) >. )
5712, 29, 39, 52cofu1 13760 . . . . . . 7  |-  ( 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 >. ) ) )
58 fvex 5541 . . . . . . . . . 10  |-  ( 1st `  Y )  e.  _V
59 fvex 5541 . . . . . . . . . . 11  |-  ( 2nd `  Y )  e.  _V
6059tposex 6270 . . . . . . . . . 10  |- tpos  ( 2nd `  Y )  e.  _V
6158, 60op1st 6130 . . . . . . . . 9  |-  ( 1st `  <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >. )  =  ( 1st `  Y
)
6261a1i 10 . . . . . . . 8  |-  ( ph  ->  ( 1st `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. )  =  ( 1st `  Y ) )
636, 12, 55, 27, 17, 28, 522ndf1 13971 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  ( 2nd `  <. F ,  X >. )
)
64 op2ndg 6135 . . . . . . . . . 10  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  ( 2nd `  <. F ,  X >. )  =  X )
6549, 50, 64syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  <. F ,  X >. )  =  X )
6663, 65eqtrd 2317 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  X )
6762, 66fveq12d 5533 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. ) `  (
( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) )  =  ( ( 1st `  Y ) `
 X ) )
6857, 67eqtrd 2317 . . . . . 6  |-  ( ph  ->  ( ( 1st `  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ) `  <. F ,  X >. )  =  ( ( 1st `  Y ) `  X
) )
696, 12, 55, 27, 17, 41, 521stf1 13968 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  ( 1st `  <. F ,  X >. )
)
70 op1stg 6134 . . . . . . . 8  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  ( 1st `  <. F ,  X >. )  =  F )
7149, 50, 70syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( 1st `  <. F ,  X >. )  =  F )
7269, 71eqtrd 2317 . . . . . 6  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  F )
7368, 72opeq12d 3806 . . . . 5  |-  ( ph  -> 
<. ( ( 1st `  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ) `  <. F ,  X >. ) ,  ( ( 1st `  ( Q  1stF  O )
) `  <. F ,  X >. ) >.  =  <. ( ( 1st `  Y
) `  X ) ,  F >. )
7456, 73eqtrd 2317 . . . 4  |-  ( ph  ->  ( ( 1st `  (
( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) `  <. F ,  X >. )  =  <. ( ( 1st `  Y ) `  X
) ,  F >. )
7574fveq2d 5531 . . 3  |-  ( ph  ->  ( ( 1st `  H
) `  ( ( 1st `  ( ( <.
( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) `  <. F ,  X >. )
)  =  ( ( 1st `  H ) `
 <. ( ( 1st `  Y ) `  X
) ,  F >. ) )
76 df-ov 5863 . . 3  |-  ( ( ( 1st `  Y
) `  X )
( 1st `  H
) F )  =  ( ( 1st `  H
) `  <. ( ( 1st `  Y ) `
 X ) ,  F >. )
7775, 76syl6eqr 2335 . 2  |-  ( ph  ->  ( ( 1st `  H
) `  ( ( 1st `  ( ( <.
( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) `  <. F ,  X >. )
)  =  ( ( ( 1st `  Y
) `  X )
( 1st `  H
) F ) )
78 eqid 2285 . . . 4  |-  ( O Nat 
S )  =  ( O Nat  S )
797, 78fuchom 13837 . . 3  |-  ( O Nat 
S )  =  (  Hom  `  Q )
8032, 10, 15, 50, 9, 24, 23, 33yon1cl 14039 . . 3  |-  ( ph  ->  ( ( 1st `  Y
) `  X )  e.  ( O  Func  S
) )
8144, 27, 8, 79, 80, 49hof1 14030 . 2  |-  ( ph  ->  ( ( ( 1st `  Y ) `  X
) ( 1st `  H
) F )  =  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )
8254, 77, 813eqtrd 2321 1  |-  ( ph  ->  ( F ( 1st `  Z ) X )  =  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1625    e. wcel 1686   _Vcvv 2790    u. cun 3152    C_ wss 3154   <.cop 3645   class class class wbr 4025    X. cxp 4689   ran crn 4692   Rel wrel 4696   ` cfv 5257  (class class class)co 5860   1stc1st 6122   2ndc2nd 6123  tpos ctpos 6235   Basecbs 13150    Hom chom 13221   Catccat 13568   Idccid 13569    Homf chomf 13570  oppCatcoppc 13616    Func cfunc 13730    o.func ccofu 13732   Nat cnat 13817   FuncCat cfuc 13818   SetCatcsetc 13909    X.c cxpc 13944    1stF c1stf 13945    2ndF c2ndf 13946   ⟨,⟩F cprf 13947   evalF cevlf 13985  HomFchof 14024  Yoncyon 14025
This theorem is referenced by:  yonedalem3a  14050  yonedalem3b  14055  yonedainv  14057  yonffthlem  14058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-tpos 6236  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-map 6776  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-fz 10785  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-hom 13234  df-cco 13235  df-cat 13572  df-cid 13573  df-homf 13574  df-comf 13575  df-oppc 13617  df-func 13734  df-cofu 13736  df-nat 13819  df-fuc 13820  df-setc 13910  df-xpc 13948  df-1stf 13949  df-2ndf 13950  df-prf 13951  df-curf 13990  df-hof 14026  df-yon 14027
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