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Theorem yonedalem21 14362
Description: Lemma for yoneda 14372. (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 5723 . . . . 5  |-  ( 1st `  Z )  =  ( 1st `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) )
32oveqi 6086 . . . 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 6076 . . . 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 2455 . . 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 2435 . . . . 5  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
7 yoneda.q . . . . . 6  |-  Q  =  ( O FuncCat  S )
87fucbas 14149 . . . . 5  |-  ( O 
Func  S )  =  (
Base `  Q )
9 yoneda.o . . . . . 6  |-  O  =  (oppCat `  C )
10 yoneda.b . . . . . 6  |-  B  =  ( Base `  C
)
119, 10oppcbas 13936 . . . . 5  |-  B  =  ( Base `  O
)
126, 8, 11xpcbas 14267 . . . 4  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
13 eqid 2435 . . . . 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 2435 . . . . 5  |-  ( (oppCat `  Q )  X.c  Q )  =  ( (oppCat `  Q )  X.c  Q )
15 yoneda.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
169oppccat 13940 . . . . . . . . 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 3517 . . . . . . . . . 10  |-  ( ph  ->  U  C_  V )
2118, 20ssexd 4342 . . . . . . . . 9  |-  ( ph  ->  U  e.  _V )
22 yoneda.s . . . . . . . . . 10  |-  S  =  ( SetCat `  U )
2322setccat 14232 . . . . . . . . 9  |-  ( U  e.  _V  ->  S  e.  Cat )
2421, 23syl 16 . . . . . . . 8  |-  ( ph  ->  S  e.  Cat )
257, 17, 24fuccat 14159 . . . . . . 7  |-  ( ph  ->  Q  e.  Cat )
26 eqid 2435 . . . . . . 7  |-  ( Q  2ndF  O )  =  ( Q  2ndF  O )
276, 25, 17, 262ndfcl 14287 . . . . . 6  |-  ( ph  ->  ( Q  2ndF  O )  e.  ( ( Q  X.c  O
)  Func  O )
)
28 eqid 2435 . . . . . . . 8  |-  (oppCat `  Q )  =  (oppCat `  Q )
29 relfunc 14051 . . . . . . . . 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 14351 . . . . . . . . 9  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
33 1st2ndbr 6388 . . . . . . . . 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 14064 . . . . . . 7  |-  ( ph  ->  ( 1st `  Y
) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y ) )
36 df-br 4205 . . . . . . 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 14077 . . . . 5  |-  ( ph  ->  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) )  e.  ( ( Q  X.c  O ) 
Func  (oppCat `  Q )
) )
39 eqid 2435 . . . . . 6  |-  ( Q  1stF  O )  =  ( Q  1stF  O )
406, 25, 17, 391stfcl 14286 . . . . 5  |-  ( ph  ->  ( Q  1stF  O )  e.  ( ( Q  X.c  O
)  Func  Q )
)
4113, 14, 38, 40prfcl 14292 . . . 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 3516 . . . . 5  |-  ( ph  ->  ran  (  Homf  `  Q ) 
C_  V )
4542, 28, 43, 25, 18, 44hofcl 14348 . . . 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 4902 . . . . 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 14073 . . 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 2479 . 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 2435 . . . . . 6  |-  (  Hom  `  ( Q  X.c  O ) )  =  (  Hom  `  ( Q  X.c  O ) )
5313, 12, 52, 38, 40, 49prf1 14289 . . . . 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 14073 . . . . . . 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 5734 . . . . . . . . . 10  |-  ( 1st `  Y )  e.  _V
56 fvex 5734 . . . . . . . . . . 11  |-  ( 2nd `  Y )  e.  _V
5756tposex 6505 . . . . . . . . . 10  |- tpos  ( 2nd `  Y )  e.  _V
5855, 57op1st 6347 . . . . . . . . 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 14284 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  ( 2nd `  <. F ,  X >. )
)
61 op2ndg 6352 . . . . . . . . . 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 2467 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  X )
6459, 63fveq12d 5726 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. ) `  (
( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) )  =  ( ( 1st `  Y ) `
 X ) )
6554, 64eqtrd 2467 . . . . . 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 14281 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  ( 1st `  <. F ,  X >. )
)
67 op1stg 6351 . . . . . . . 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 2467 . . . . . 6  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  F )
7065, 69opeq12d 3984 . . . . 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 2467 . . . 4  |-  ( ph  ->  ( ( 1st `  (
( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) `  <. F ,  X >. )  =  <. ( ( 1st `  Y ) `  X
) ,  F >. )
7271fveq2d 5724 . . 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 6076 . . 3  |-  ( ( ( 1st `  Y
) `  X )
( 1st `  H
) F )  =  ( ( 1st `  H
) `  <. ( ( 1st `  Y ) `
 X ) ,  F >. )
7472, 73syl6eqr 2485 . 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 2435 . . . 4  |-  ( O Nat 
S )  =  ( O Nat  S )
767, 75fuchom 14150 . . 3  |-  ( O Nat 
S )  =  (  Hom  `  Q )
7730, 10, 15, 47, 9, 22, 21, 31yon1cl 14352 . . 3  |-  ( ph  ->  ( ( 1st `  Y
) `  X )  e.  ( O  Func  S
) )
7842, 25, 8, 76, 77, 46hof1 14343 . 2  |-  ( ph  ->  ( ( ( 1st `  Y ) `  X
) ( 1st `  H
) F )  =  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )
7951, 74, 783eqtrd 2471 1  |-  ( ph  ->  ( F ( 1st `  Z ) X )  =  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948    u. cun 3310    C_ wss 3312   <.cop 3809   class class class wbr 4204    X. cxp 4868   ran crn 4871   Rel wrel 4875   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340  tpos ctpos 6470   Basecbs 13461    Hom chom 13532   Catccat 13881   Idccid 13882    Homf chomf 13883  oppCatcoppc 13929    Func cfunc 14043    o.func ccofu 14045   Nat cnat 14130   FuncCat cfuc 14131   SetCatcsetc 14222    X.c cxpc 14257    1stF c1stf 14258    2ndF c2ndf 14259   ⟨,⟩F cprf 14260   evalF cevlf 14298  HomFchof 14337  Yoncyon 14338
This theorem is referenced by:  yonedalem3a  14363  yonedalem3b  14368  yonedainv  14370  yonffthlem  14371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-hom 13545  df-cco 13546  df-cat 13885  df-cid 13886  df-homf 13887  df-comf 13888  df-oppc 13930  df-func 14047  df-cofu 14049  df-nat 14132  df-fuc 14133  df-setc 14223  df-xpc 14261  df-1stf 14262  df-2ndf 14263  df-prf 14264  df-curf 14303  df-hof 14339  df-yon 14340
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