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Theorem yonedalem21 14375
Description: Lemma for yoneda 14385. (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 5734 . . . . 5  |-  ( 1st `  Z )  =  ( 1st `  ( H  o.func  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) ) ) )
32oveqi 6097 . . . 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 6087 . . . 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 2458 . . 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 2438 . . . . 5  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
7 yoneda.q . . . . . 6  |-  Q  =  ( O FuncCat  S )
87fucbas 14162 . . . . 5  |-  ( O 
Func  S )  =  (
Base `  Q )
9 yoneda.o . . . . . 6  |-  O  =  (oppCat `  C )
10 yoneda.b . . . . . 6  |-  B  =  ( Base `  C
)
119, 10oppcbas 13949 . . . . 5  |-  B  =  ( Base `  O
)
126, 8, 11xpcbas 14280 . . . 4  |-  ( ( O  Func  S )  X.  B )  =  (
Base `  ( Q  X.c  O ) )
13 eqid 2438 . . . . 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 2438 . . . . 5  |-  ( (oppCat `  Q )  X.c  Q )  =  ( (oppCat `  Q )  X.c  Q )
15 yoneda.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
169oppccat 13953 . . . . . . . . 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 3527 . . . . . . . . . 10  |-  ( ph  ->  U  C_  V )
2118, 20ssexd 4353 . . . . . . . . 9  |-  ( ph  ->  U  e.  _V )
22 yoneda.s . . . . . . . . . 10  |-  S  =  ( SetCat `  U )
2322setccat 14245 . . . . . . . . 9  |-  ( U  e.  _V  ->  S  e.  Cat )
2421, 23syl 16 . . . . . . . 8  |-  ( ph  ->  S  e.  Cat )
257, 17, 24fuccat 14172 . . . . . . 7  |-  ( ph  ->  Q  e.  Cat )
26 eqid 2438 . . . . . . 7  |-  ( Q  2ndF  O )  =  ( Q  2ndF  O )
276, 25, 17, 262ndfcl 14300 . . . . . 6  |-  ( ph  ->  ( Q  2ndF  O )  e.  ( ( Q  X.c  O
)  Func  O )
)
28 eqid 2438 . . . . . . . 8  |-  (oppCat `  Q )  =  (oppCat `  Q )
29 relfunc 14064 . . . . . . . . 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 14364 . . . . . . . . 9  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
33 1st2ndbr 6399 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  ( 1st `  Y ) ( C  Func  Q )
( 2nd `  Y
) )
3429, 32, 33sylancr 646 . . . . . . . 8  |-  ( ph  ->  ( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
359, 28, 34funcoppc 14077 . . . . . . 7  |-  ( ph  ->  ( 1st `  Y
) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y ) )
36 df-br 4216 . . . . . . 7  |-  ( ( 1st `  Y ) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y )  <->  <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  e.  ( O  Func  (oppCat `  Q ) ) )
3735, 36sylib 190 . . . . . 6  |-  ( ph  -> 
<. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  e.  ( O  Func  (oppCat `  Q
) ) )
3827, 37cofucl 14090 . . . . 5  |-  ( ph  ->  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) )  e.  ( ( Q  X.c  O ) 
Func  (oppCat `  Q )
) )
39 eqid 2438 . . . . . 6  |-  ( Q  1stF  O )  =  ( Q  1stF  O )
406, 25, 17, 391stfcl 14299 . . . . 5  |-  ( ph  ->  ( Q  1stF  O )  e.  ( ( Q  X.c  O
)  Func  Q )
)
4113, 14, 38, 40prfcl 14305 . . . 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 3526 . . . . 5  |-  ( ph  ->  ran  (  Homf  `  Q ) 
C_  V )
4542, 28, 43, 25, 18, 44hofcl 14361 . . . 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 4913 . . . . 5  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  <. F ,  X >.  e.  ( ( O  Func  S )  X.  B ) )
4946, 47, 48syl2anc 644 . . . 4  |-  ( ph  -> 
<. F ,  X >.  e.  ( ( O  Func  S )  X.  B ) )
5012, 41, 45, 49cofu1 14086 . . 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 2482 . 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 2438 . . . . . 6  |-  (  Hom  `  ( Q  X.c  O ) )  =  (  Hom  `  ( Q  X.c  O ) )
5313, 12, 52, 38, 40, 49prf1 14302 . . . . 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 14086 . . . . . . 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 5745 . . . . . . . . . 10  |-  ( 1st `  Y )  e.  _V
56 fvex 5745 . . . . . . . . . . 11  |-  ( 2nd `  Y )  e.  _V
5756tposex 6516 . . . . . . . . . 10  |- tpos  ( 2nd `  Y )  e.  _V
5855, 57op1st 6358 . . . . . . . . 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 14297 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  ( 2nd `  <. F ,  X >. )
)
61 op2ndg 6363 . . . . . . . . . 10  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  ( 2nd `  <. F ,  X >. )  =  X )
6246, 47, 61syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  <. F ,  X >. )  =  X )
6360, 62eqtrd 2470 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. )  =  X )
6459, 63fveq12d 5737 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >. ) `  (
( 1st `  ( Q  2ndF  O ) ) `  <. F ,  X >. ) )  =  ( ( 1st `  Y ) `
 X ) )
6554, 64eqtrd 2470 . . . . . 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 14294 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  ( 1st `  <. F ,  X >. )
)
67 op1stg 6362 . . . . . . . 8  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  ( 1st `  <. F ,  X >. )  =  F )
6846, 47, 67syl2anc 644 . . . . . . 7  |-  ( ph  ->  ( 1st `  <. F ,  X >. )  =  F )
6966, 68eqtrd 2470 . . . . . 6  |-  ( ph  ->  ( ( 1st `  ( Q  1stF  O ) ) `  <. F ,  X >. )  =  F )
7065, 69opeq12d 3994 . . . . 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 2470 . . . 4  |-  ( ph  ->  ( ( 1st `  (
( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) ) `  <. F ,  X >. )  =  <. ( ( 1st `  Y ) `  X
) ,  F >. )
7271fveq2d 5735 . . 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 6087 . . 3  |-  ( ( ( 1st `  Y
) `  X )
( 1st `  H
) F )  =  ( ( 1st `  H
) `  <. ( ( 1st `  Y ) `
 X ) ,  F >. )
7472, 73syl6eqr 2488 . 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 2438 . . . 4  |-  ( O Nat 
S )  =  ( O Nat  S )
767, 75fuchom 14163 . . 3  |-  ( O Nat 
S )  =  (  Hom  `  Q )
7730, 10, 15, 47, 9, 22, 21, 31yon1cl 14365 . . 3  |-  ( ph  ->  ( ( 1st `  Y
) `  X )  e.  ( O  Func  S
) )
7842, 25, 8, 76, 77, 46hof1 14356 . 2  |-  ( ph  ->  ( ( ( 1st `  Y ) `  X
) ( 1st `  H
) F )  =  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )
7951, 74, 783eqtrd 2474 1  |-  ( ph  ->  ( F ( 1st `  Z ) X )  =  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958    u. cun 3320    C_ wss 3322   <.cop 3819   class class class wbr 4215    X. cxp 4879   ran crn 4882   Rel wrel 4886   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351  tpos ctpos 6481   Basecbs 13474    Hom chom 13545   Catccat 13894   Idccid 13895    Homf chomf 13896  oppCatcoppc 13942    Func cfunc 14056    o.func ccofu 14058   Nat cnat 14143   FuncCat cfuc 14144   SetCatcsetc 14235    X.c cxpc 14270    1stF c1stf 14271    2ndF c2ndf 14272   ⟨,⟩F cprf 14273   evalF cevlf 14311  HomFchof 14350  Yoncyon 14351
This theorem is referenced by:  yonedalem3a  14376  yonedalem3b  14381  yonedainv  14383  yonffthlem  14384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-tpos 6482  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-fz 11049  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-hom 13558  df-cco 13559  df-cat 13898  df-cid 13899  df-homf 13900  df-comf 13901  df-oppc 13943  df-func 14060  df-cofu 14062  df-nat 14145  df-fuc 14146  df-setc 14236  df-xpc 14274  df-1stf 14275  df-2ndf 14276  df-prf 14277  df-curf 14316  df-hof 14352  df-yon 14353
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