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Theorem yonedalem21 14298
Description: Lemma for yoneda 14308. (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 5672 . . . . 5  |-  ( 1st `  Z )  =  ( 1st `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) )
32oveqi 6034 . . . 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 6024 . . . 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 2408 . . 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 2388 . . . . 5  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
7 yoneda.q . . . . . 6  |-  Q  =  ( O FuncCat  S )
87fucbas 14085 . . . . 5  |-  ( O 
Func  S )  =  (
Base `  Q )
9 yoneda.o . . . . . 6  |-  O  =  (oppCat `  C )
10 yoneda.b . . . . . 6  |-  B  =  ( Base `  C
)
119, 10oppcbas 13872 . . . . 5  |-  B  =  ( Base `  O
)
126, 8, 11xpcbas 14203 . . . 4  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
13 eqid 2388 . . . . 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 2388 . . . . 5  |-  ( (oppCat `  Q )  X.c  Q )  =  ( (oppCat `  Q )  X.c  Q )
15 yoneda.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
169oppccat 13876 . . . . . . . . 9  |-  ( C  e.  Cat  ->  O  e.  Cat )
1715, 16syl 16 . . . . . . . 8  |-  ( ph  ->  O  e.  Cat )
18 yoneda.w . . . . . . . . . 10  |-  ( ph  ->  V  e.  W )
19 yoneda.v . . . . . . . . . . 11  |-  ( ph  ->  ( ran  (  Homf  `  Q )  u.  U
)  C_  V )
2019unssbd 3469 . . . . . . . . . 10  |-  ( ph  ->  U  C_  V )
2118, 20ssexd 4292 . . . . . . . . 9  |-  ( ph  ->  U  e.  _V )
22 yoneda.s . . . . . . . . . 10  |-  S  =  ( SetCat `  U )
2322setccat 14168 . . . . . . . . 9  |-  ( U  e.  _V  ->  S  e.  Cat )
2421, 23syl 16 . . . . . . . 8  |-  ( ph  ->  S  e.  Cat )
257, 17, 24fuccat 14095 . . . . . . 7  |-  ( ph  ->  Q  e.  Cat )
26 eqid 2388 . . . . . . 7  |-  ( Q  2ndF  O )  =  ( Q  2ndF  O )
276, 25, 17, 262ndfcl 14223 . . . . . 6  |-  ( ph  ->  ( Q  2ndF  O )  e.  ( ( Q  X.c  O
)  Func  O )
)
28 eqid 2388 . . . . . . . 8  |-  (oppCat `  Q )  =  (oppCat `  Q )
29 relfunc 13987 . . . . . . . . 9  |-  Rel  ( C  Func  Q )
30 yoneda.y . . . . . . . . . 10  |-  Y  =  (Yon `  C )
31 yoneda.u . . . . . . . . . 10  |-  ( ph  ->  ran  (  Homf  `  C ) 
C_  U )
3230, 15, 9, 22, 7, 21, 31yoncl 14287 . . . . . . . . 9  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
33 1st2ndbr 6336 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  ( 1st `  Y ) ( C  Func  Q )
( 2nd `  Y
) )
3429, 32, 33sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
359, 28, 34funcoppc 14000 . . . . . . 7  |-  ( ph  ->  ( 1st `  Y
) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y ) )
36 df-br 4155 . . . . . . 7  |-  ( ( 1st `  Y ) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y )  <->  <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  e.  ( O  Func  (oppCat `  Q ) ) )
3735, 36sylib 189 . . . . . 6  |-  ( ph  -> 
<. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  e.  ( O  Func  (oppCat `  Q
) ) )
3827, 37cofucl 14013 . . . . 5  |-  ( ph  ->  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) )  e.  ( ( Q  X.c  O ) 
Func  (oppCat `  Q )
) )
39 eqid 2388 . . . . . 6  |-  ( Q  1stF  O )  =  ( Q  1stF  O )
406, 25, 17, 391stfcl 14222 . . . . 5  |-  ( ph  ->  ( Q  1stF  O )  e.  ( ( Q  X.c  O
)  Func  Q )
)
4113, 14, 38, 40prfcl 14228 . . . 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 ) ) )
42 yoneda.h . . . . 5  |-  H  =  (HomF
`  Q )
43 yoneda.t . . . . 5  |-  T  =  ( SetCat `  V )
4419unssad 3468 . . . . 5  |-  ( ph  ->  ran  (  Homf  `  Q ) 
C_  V )
4542, 28, 43, 25, 18, 44hofcl 14284 . . . 4  |-  ( ph  ->  H  e.  ( ( (oppCat `  Q )  X.c  Q )  Func  T
) )
46 yonedalem21.f . . . . 5  |-  ( ph  ->  F  e.  ( O 
Func  S ) )
47 yonedalem21.x . . . . 5  |-  ( ph  ->  X  e.  B )
48 opelxpi 4851 . . . . 5  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  <. F ,  X >.  e.  ( ( O  Func  S )  X.  B ) )
4946, 47, 48syl2anc 643 . . . 4  |-  ( ph  -> 
<. F ,  X >.  e.  ( ( O  Func  S )  X.  B ) )
5012, 41, 45, 49cofu1 14009 . . 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 >. )
) )
515, 50syl5eq 2432 . 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 >. )
) )
52 eqid 2388 . . . . . 6  |-  (  Hom  `  ( Q  X.c  O ) )  =  (  Hom  `  ( Q  X.c  O ) )
5313, 12, 52, 38, 40, 49prf1 14225 . . . . 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 >. ) >. )
5412, 27, 37, 49cofu1 14009 . . . . . . 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 >. ) ) )
55 fvex 5683 . . . . . . . . . 10  |-  ( 1st `  Y )  e.  _V
56 fvex 5683 . . . . . . . . . . 11  |-  ( 2nd `  Y )  e.  _V
5756tposex 6450 . . . . . . . . . 10  |- tpos  ( 2nd `  Y )  e.  _V
5855, 57op1st 6295 . . . . . . . . 9  |-  ( 1st `  <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >. )  =  ( 1st `  Y
)
5958a1i 11 . . . . . . . 8  |-  ( ph  ->  ( 1st `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. )  =  ( 1st `  Y ) )
606, 12, 52, 25, 17, 26, 492ndf1 14220 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  ( 2nd `  <. F ,  X >. )
)
61 op2ndg 6300 . . . . . . . . . 10  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  ( 2nd `  <. F ,  X >. )  =  X )
6246, 47, 61syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  <. F ,  X >. )  =  X )
6360, 62eqtrd 2420 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  X )
6459, 63fveq12d 5675 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. ) `  (
( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) )  =  ( ( 1st `  Y ) `
 X ) )
6554, 64eqtrd 2420 . . . . . 6  |-  ( ph  ->  ( ( 1st `  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ) `  <. F ,  X >. )  =  ( ( 1st `  Y ) `  X
) )
666, 12, 52, 25, 17, 39, 491stf1 14217 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  ( 1st `  <. F ,  X >. )
)
67 op1stg 6299 . . . . . . . 8  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  ( 1st `  <. F ,  X >. )  =  F )
6846, 47, 67syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( 1st `  <. F ,  X >. )  =  F )
6966, 68eqtrd 2420 . . . . . 6  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  F )
7065, 69opeq12d 3935 . . . . 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 >. )
7153, 70eqtrd 2420 . . . 4  |-  ( ph  ->  ( ( 1st `  (
( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) `  <. F ,  X >. )  =  <. ( ( 1st `  Y ) `  X
) ,  F >. )
7271fveq2d 5673 . . 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 >. ) )
73 df-ov 6024 . . 3  |-  ( ( ( 1st `  Y
) `  X )
( 1st `  H
) F )  =  ( ( 1st `  H
) `  <. ( ( 1st `  Y ) `
 X ) ,  F >. )
7472, 73syl6eqr 2438 . 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 ) )
75 eqid 2388 . . . 4  |-  ( O Nat 
S )  =  ( O Nat  S )
767, 75fuchom 14086 . . 3  |-  ( O Nat 
S )  =  (  Hom  `  Q )
7730, 10, 15, 47, 9, 22, 21, 31yon1cl 14288 . . 3  |-  ( ph  ->  ( ( 1st `  Y
) `  X )  e.  ( O  Func  S
) )
7842, 25, 8, 76, 77, 46hof1 14279 . 2  |-  ( ph  ->  ( ( ( 1st `  Y ) `  X
) ( 1st `  H
) F )  =  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )
7951, 74, 783eqtrd 2424 1  |-  ( ph  ->  ( F ( 1st `  Z ) X )  =  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2900    u. cun 3262    C_ wss 3264   <.cop 3761   class class class wbr 4154    X. cxp 4817   ran crn 4820   Rel wrel 4824   ` cfv 5395  (class class class)co 6021   1stc1st 6287   2ndc2nd 6288  tpos ctpos 6415   Basecbs 13397    Hom chom 13468   Catccat 13817   Idccid 13818    Homf chomf 13819  oppCatcoppc 13865    Func cfunc 13979    o.func ccofu 13981   Nat cnat 14066   FuncCat cfuc 14067   SetCatcsetc 14158    X.c cxpc 14193    1stF c1stf 14194    2ndF c2ndf 14195   ⟨,⟩F cprf 14196   evalF cevlf 14234  HomFchof 14273  Yoncyon 14274
This theorem is referenced by:  yonedalem3a  14299  yonedalem3b  14304  yonedainv  14306  yonffthlem  14307
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-tpos 6416  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-fz 10977  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-hom 13481  df-cco 13482  df-cat 13821  df-cid 13822  df-homf 13823  df-comf 13824  df-oppc 13866  df-func 13983  df-cofu 13985  df-nat 14068  df-fuc 14069  df-setc 14159  df-xpc 14197  df-1stf 14198  df-2ndf 14199  df-prf 14200  df-curf 14239  df-hof 14275  df-yon 14276
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