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Theorem yonedalem21 15399
Description: Lemma for yoneda 15409. (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  ( 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 )
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 5868 . . . . 5  |-  ( 1st `  Z )  =  ( 1st `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) )
32oveqi 6296 . . . 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 6286 . . . 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 2496 . . 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 2467 . . . . 5  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
7 yoneda.q . . . . . 6  |-  Q  =  ( O FuncCat  S )
87fucbas 15186 . . . . 5  |-  ( O 
Func  S )  =  (
Base `  Q )
9 yoneda.o . . . . . 6  |-  O  =  (oppCat `  C )
10 yoneda.b . . . . . 6  |-  B  =  ( Base `  C
)
119, 10oppcbas 14973 . . . . 5  |-  B  =  ( Base `  O
)
126, 8, 11xpcbas 15304 . . . 4  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
13 eqid 2467 . . . . 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 2467 . . . . 5  |-  ( (oppCat `  Q )  X.c  Q )  =  ( (oppCat `  Q )  X.c  Q )
15 yoneda.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
169oppccat 14977 . . . . . . . . 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  ( Hom f  `  Q )  u.  U
)  C_  V )
2019unssbd 3682 . . . . . . . . . 10  |-  ( ph  ->  U  C_  V )
2118, 20ssexd 4594 . . . . . . . . 9  |-  ( ph  ->  U  e.  _V )
22 yoneda.s . . . . . . . . . 10  |-  S  =  ( SetCat `  U )
2322setccat 15269 . . . . . . . . 9  |-  ( U  e.  _V  ->  S  e.  Cat )
2421, 23syl 16 . . . . . . . 8  |-  ( ph  ->  S  e.  Cat )
257, 17, 24fuccat 15196 . . . . . . 7  |-  ( ph  ->  Q  e.  Cat )
26 eqid 2467 . . . . . . 7  |-  ( Q  2ndF  O )  =  ( Q  2ndF  O )
276, 25, 17, 262ndfcl 15324 . . . . . 6  |-  ( ph  ->  ( Q  2ndF  O )  e.  ( ( Q  X.c  O
)  Func  O )
)
28 eqid 2467 . . . . . . . 8  |-  (oppCat `  Q )  =  (oppCat `  Q )
29 relfunc 15088 . . . . . . . . 9  |-  Rel  ( C  Func  Q )
30 yoneda.y . . . . . . . . . 10  |-  Y  =  (Yon `  C )
31 yoneda.u . . . . . . . . . 10  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
3230, 15, 9, 22, 7, 21, 31yoncl 15388 . . . . . . . . 9  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
33 1st2ndbr 6833 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  ( 1st `  Y ) ( C  Func  Q )
( 2nd `  Y
) )
3429, 32, 33sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
359, 28, 34funcoppc 15101 . . . . . . 7  |-  ( ph  ->  ( 1st `  Y
) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y ) )
36 df-br 4448 . . . . . . 7  |-  ( ( 1st `  Y ) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y )  <->  <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  e.  ( O  Func  (oppCat `  Q ) ) )
3735, 36sylib 196 . . . . . 6  |-  ( ph  -> 
<. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  e.  ( O  Func  (oppCat `  Q
) ) )
3827, 37cofucl 15114 . . . . 5  |-  ( ph  ->  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) )  e.  ( ( Q  X.c  O ) 
Func  (oppCat `  Q )
) )
39 eqid 2467 . . . . . 6  |-  ( Q  1stF  O )  =  ( Q  1stF  O )
406, 25, 17, 391stfcl 15323 . . . . 5  |-  ( ph  ->  ( Q  1stF  O )  e.  ( ( Q  X.c  O
)  Func  Q )
)
4113, 14, 38, 40prfcl 15329 . . . 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 3681 . . . . 5  |-  ( ph  ->  ran  ( Hom f  `  Q ) 
C_  V )
4542, 28, 43, 25, 18, 44hofcl 15385 . . . 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 5030 . . . . 5  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  <. F ,  X >.  e.  ( ( O  Func  S )  X.  B ) )
4946, 47, 48syl2anc 661 . . . 4  |-  ( ph  -> 
<. F ,  X >.  e.  ( ( O  Func  S )  X.  B ) )
5012, 41, 45, 49cofu1 15110 . . 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 2520 . 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 2467 . . . . . 6  |-  ( Hom  `  ( Q  X.c  O ) )  =  ( Hom  `  ( Q  X.c  O ) )
5313, 12, 52, 38, 40, 49prf1 15326 . . . . 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 15110 . . . . . . 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 5875 . . . . . . . . . 10  |-  ( 1st `  Y )  e.  _V
56 fvex 5875 . . . . . . . . . . 11  |-  ( 2nd `  Y )  e.  _V
5756tposex 6989 . . . . . . . . . 10  |- tpos  ( 2nd `  Y )  e.  _V
5855, 57op1st 6792 . . . . . . . . 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 15321 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  ( 2nd `  <. F ,  X >. )
)
61 op2ndg 6797 . . . . . . . . . 10  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  ( 2nd `  <. F ,  X >. )  =  X )
6246, 47, 61syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  <. F ,  X >. )  =  X )
6360, 62eqtrd 2508 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  X )
6459, 63fveq12d 5871 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. ) `  (
( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) )  =  ( ( 1st `  Y ) `
 X ) )
6554, 64eqtrd 2508 . . . . . 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 15318 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  ( 1st `  <. F ,  X >. )
)
67 op1stg 6796 . . . . . . . 8  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  ( 1st `  <. F ,  X >. )  =  F )
6846, 47, 67syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( 1st `  <. F ,  X >. )  =  F )
6966, 68eqtrd 2508 . . . . . 6  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  F )
7065, 69opeq12d 4221 . . . . 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 2508 . . . 4  |-  ( ph  ->  ( ( 1st `  (
( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) `  <. F ,  X >. )  =  <. ( ( 1st `  Y ) `  X
) ,  F >. )
7271fveq2d 5869 . . 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 6286 . . 3  |-  ( ( ( 1st `  Y
) `  X )
( 1st `  H
) F )  =  ( ( 1st `  H
) `  <. ( ( 1st `  Y ) `
 X ) ,  F >. )
7472, 73syl6eqr 2526 . 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 2467 . . . 4  |-  ( O Nat 
S )  =  ( O Nat  S )
767, 75fuchom 15187 . . 3  |-  ( O Nat 
S )  =  ( Hom  `  Q )
7730, 10, 15, 47, 9, 22, 21, 31yon1cl 15389 . . 3  |-  ( ph  ->  ( ( 1st `  Y
) `  X )  e.  ( O  Func  S
) )
7842, 25, 8, 76, 77, 46hof1 15380 . 2  |-  ( ph  ->  ( ( ( 1st `  Y ) `  X
) ( 1st `  H
) F )  =  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )
7951, 74, 783eqtrd 2512 1  |-  ( ph  ->  ( F ( 1st `  Z ) X )  =  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113    u. cun 3474    C_ wss 3476   <.cop 4033   class class class wbr 4447    X. cxp 4997   ran crn 5000   Rel wrel 5004   ` cfv 5587  (class class class)co 6283   1stc1st 6782   2ndc2nd 6783  tpos ctpos 6954   Basecbs 14489   Hom chom 14565   Catccat 14918   Idccid 14919   Hom f chomf 14920  oppCatcoppc 14966    Func cfunc 15080    o.func ccofu 15082   Nat cnat 15167   FuncCat cfuc 15168   SetCatcsetc 15259    X.c cxpc 15294    1stF c1stf 15295    2ndF c2ndf 15296   ⟨,⟩F cprf 15297   evalF cevlf 15335  HomFchof 15374  Yoncyon 15375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-7 10598  df-8 10599  df-9 10600  df-10 10601  df-n0 10795  df-z 10864  df-dec 10976  df-uz 11082  df-fz 11672  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-hom 14578  df-cco 14579  df-cat 14922  df-cid 14923  df-homf 14924  df-comf 14925  df-oppc 14967  df-func 15084  df-cofu 15086  df-nat 15169  df-fuc 15170  df-setc 15260  df-xpc 15298  df-1stf 15299  df-2ndf 15300  df-prf 15301  df-curf 15340  df-hof 15376  df-yon 15377
This theorem is referenced by:  yonedalem3a  15400  yonedalem3b  15405  yonedainv  15407  yonffthlem  15408
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