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Theorem yon2 15396
Description: Value of the Yoneda embedding at a morphism. (Contributed by Mario Carneiro, 17-Jan-2017.)
Hypotheses
Ref Expression
yon11.y  |-  Y  =  (Yon `  C )
yon11.b  |-  B  =  ( Base `  C
)
yon11.c  |-  ( ph  ->  C  e.  Cat )
yon11.p  |-  ( ph  ->  X  e.  B )
yon11.h  |-  H  =  ( Hom  `  C
)
yon11.z  |-  ( ph  ->  Z  e.  B )
yon12.x  |-  .x.  =  (comp `  C )
yon12.w  |-  ( ph  ->  W  e.  B )
yon2.f  |-  ( ph  ->  F  e.  ( X H Z ) )
yon2.g  |-  ( ph  ->  G  e.  ( W H X ) )
Assertion
Ref Expression
yon2  |-  ( ph  ->  ( ( ( ( X ( 2nd `  Y
) Z ) `  F ) `  W
) `  G )  =  ( F (
<. W ,  X >.  .x. 
Z ) G ) )

Proof of Theorem yon2
StepHypRef Expression
1 yon11.y . . . . . . . . 9  |-  Y  =  (Yon `  C )
2 yon11.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
3 eqid 2467 . . . . . . . . 9  |-  (oppCat `  C )  =  (oppCat `  C )
4 eqid 2467 . . . . . . . . 9  |-  (HomF `  (oppCat `  C ) )  =  (HomF
`  (oppCat `  C )
)
51, 2, 3, 4yonval 15391 . . . . . . . 8  |-  ( ph  ->  Y  =  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) ) )
65fveq2d 5870 . . . . . . 7  |-  ( ph  ->  ( 2nd `  Y
)  =  ( 2nd `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) )
76oveqd 6302 . . . . . 6  |-  ( ph  ->  ( X ( 2nd `  Y ) Z )  =  ( X ( 2nd `  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) ) ) Z ) )
87fveq1d 5868 . . . . 5  |-  ( ph  ->  ( ( X ( 2nd `  Y ) Z ) `  F
)  =  ( ( X ( 2nd `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) Z ) `  F ) )
98fveq1d 5868 . . . 4  |-  ( ph  ->  ( ( ( X ( 2nd `  Y
) Z ) `  F ) `  W
)  =  ( ( ( X ( 2nd `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) Z ) `  F ) `
 W ) )
10 eqid 2467 . . . . 5  |-  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) )  =  (
<. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) )
11 yon11.b . . . . 5  |-  B  =  ( Base `  C
)
123oppccat 14981 . . . . . 6  |-  ( C  e.  Cat  ->  (oppCat `  C )  e.  Cat )
132, 12syl 16 . . . . 5  |-  ( ph  ->  (oppCat `  C )  e.  Cat )
14 eqid 2467 . . . . . 6  |-  ( SetCat ` 
ran  ( Hom f  `  C ) )  =  ( SetCat ` 
ran  ( Hom f  `  C ) )
15 fvex 5876 . . . . . . . 8  |-  ( Hom f  `  C )  e.  _V
1615rnex 6719 . . . . . . 7  |-  ran  ( Hom f  `  C )  e.  _V
1716a1i 11 . . . . . 6  |-  ( ph  ->  ran  ( Hom f  `  C )  e.  _V )
18 ssid 3523 . . . . . . 7  |-  ran  ( Hom f  `  C )  C_  ran  ( Hom f  `  C )
1918a1i 11 . . . . . 6  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  ran  ( Hom f  `  C
) )
203, 4, 14, 2, 17, 19oppchofcl 15390 . . . . 5  |-  ( ph  ->  (HomF
`  (oppCat `  C )
)  e.  ( ( C  X.c  (oppCat `  C )
)  Func  ( SetCat ` 
ran  ( Hom f  `  C ) ) ) )
213, 11oppcbas 14977 . . . . 5  |-  B  =  ( Base `  (oppCat `  C ) )
22 yon11.h . . . . 5  |-  H  =  ( Hom  `  C
)
23 eqid 2467 . . . . 5  |-  ( Id
`  (oppCat `  C )
)  =  ( Id
`  (oppCat `  C )
)
24 yon11.p . . . . 5  |-  ( ph  ->  X  e.  B )
25 yon11.z . . . . 5  |-  ( ph  ->  Z  e.  B )
26 yon2.f . . . . 5  |-  ( ph  ->  F  e.  ( X H Z ) )
27 eqid 2467 . . . . 5  |-  ( ( X ( 2nd `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) Z ) `  F )  =  ( ( X ( 2nd `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) Z ) `  F )
28 yon12.w . . . . 5  |-  ( ph  ->  W  e.  B )
2910, 11, 2, 13, 20, 21, 22, 23, 24, 25, 26, 27, 28curf2val 15360 . . . 4  |-  ( ph  ->  ( ( ( X ( 2nd `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) Z ) `  F ) `
 W )  =  ( F ( <. X ,  W >. ( 2nd `  (HomF `  (oppCat `  C ) ) )
<. Z ,  W >. ) ( ( Id `  (oppCat `  C ) ) `
 W ) ) )
309, 29eqtrd 2508 . . 3  |-  ( ph  ->  ( ( ( X ( 2nd `  Y
) Z ) `  F ) `  W
)  =  ( F ( <. X ,  W >. ( 2nd `  (HomF `  (oppCat `  C ) ) )
<. Z ,  W >. ) ( ( Id `  (oppCat `  C ) ) `
 W ) ) )
3130fveq1d 5868 . 2  |-  ( ph  ->  ( ( ( ( X ( 2nd `  Y
) Z ) `  F ) `  W
) `  G )  =  ( ( F ( <. X ,  W >. ( 2nd `  (HomF `  (oppCat `  C ) ) )
<. Z ,  W >. ) ( ( Id `  (oppCat `  C ) ) `
 W ) ) `
 G ) )
32 eqid 2467 . . 3  |-  ( Hom  `  (oppCat `  C )
)  =  ( Hom  `  (oppCat `  C )
)
33 eqid 2467 . . 3  |-  (comp `  (oppCat `  C ) )  =  (comp `  (oppCat `  C ) )
3422, 3oppchom 14974 . . . 4  |-  ( Z ( Hom  `  (oppCat `  C ) ) X )  =  ( X H Z )
3526, 34syl6eleqr 2566 . . 3  |-  ( ph  ->  F  e.  ( Z ( Hom  `  (oppCat `  C ) ) X ) )
3621, 32, 23, 13, 28catidcl 14940 . . 3  |-  ( ph  ->  ( ( Id `  (oppCat `  C ) ) `
 W )  e.  ( W ( Hom  `  (oppCat `  C )
) W ) )
37 yon2.g . . . 4  |-  ( ph  ->  G  e.  ( W H X ) )
3822, 3oppchom 14974 . . . 4  |-  ( X ( Hom  `  (oppCat `  C ) ) W )  =  ( W H X )
3937, 38syl6eleqr 2566 . . 3  |-  ( ph  ->  G  e.  ( X ( Hom  `  (oppCat `  C ) ) W ) )
404, 13, 21, 32, 24, 28, 25, 28, 33, 35, 36, 39hof2 15387 . 2  |-  ( ph  ->  ( ( F (
<. X ,  W >. ( 2nd `  (HomF `  (oppCat `  C ) ) )
<. Z ,  W >. ) ( ( Id `  (oppCat `  C ) ) `
 W ) ) `
 G )  =  ( ( ( ( Id `  (oppCat `  C ) ) `  W ) ( <. X ,  W >. (comp `  (oppCat `  C )
) W ) G ) ( <. Z ,  X >. (comp `  (oppCat `  C ) ) W ) F ) )
4121, 32, 23, 13, 24, 33, 28, 39catlid 14941 . . . 4  |-  ( ph  ->  ( ( ( Id
`  (oppCat `  C )
) `  W )
( <. X ,  W >. (comp `  (oppCat `  C
) ) W ) G )  =  G )
4241oveq1d 6300 . . 3  |-  ( ph  ->  ( ( ( ( Id `  (oppCat `  C ) ) `  W ) ( <. X ,  W >. (comp `  (oppCat `  C )
) W ) G ) ( <. Z ,  X >. (comp `  (oppCat `  C ) ) W ) F )  =  ( G ( <. Z ,  X >. (comp `  (oppCat `  C )
) W ) F ) )
43 yon12.x . . . 4  |-  .x.  =  (comp `  C )
4411, 43, 3, 25, 24, 28oppcco 14976 . . 3  |-  ( ph  ->  ( G ( <. Z ,  X >. (comp `  (oppCat `  C )
) W ) F )  =  ( F ( <. W ,  X >.  .x.  Z ) G ) )
4542, 44eqtrd 2508 . 2  |-  ( ph  ->  ( ( ( ( Id `  (oppCat `  C ) ) `  W ) ( <. X ,  W >. (comp `  (oppCat `  C )
) W ) G ) ( <. Z ,  X >. (comp `  (oppCat `  C ) ) W ) F )  =  ( F ( <. W ,  X >.  .x. 
Z ) G ) )
4631, 40, 453eqtrd 2512 1  |-  ( ph  ->  ( ( ( ( X ( 2nd `  Y
) Z ) `  F ) `  W
) `  G )  =  ( F (
<. W ,  X >.  .x. 
Z ) G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   <.cop 4033   ran crn 5000   ` cfv 5588  (class class class)co 6285   2ndc2nd 6784   Basecbs 14493   Hom chom 14569  compcco 14570   Catccat 14922   Idccid 14923   Hom f chomf 14924  oppCatcoppc 14970   SetCatcsetc 15263   curryF ccurf 15340  HomFchof 15378  Yoncyon 15379
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 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6956  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-uz 11084  df-fz 11674  df-struct 14495  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-hom 14582  df-cco 14583  df-cat 14926  df-cid 14927  df-homf 14928  df-comf 14929  df-oppc 14971  df-func 15088  df-setc 15264  df-xpc 15302  df-curf 15344  df-hof 15380  df-yon 15381
This theorem is referenced by:  yonedalem3b  15409  yonffthlem  15412
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