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Theorem yon12 15408
Description: Value of the Yoneda embedding at a morphism. The partially evaluated Yoneda embedding is also the contravariant Hom functor. (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 )
yon12.f  |-  ( ph  ->  F  e.  ( W H Z ) )
yon12.g  |-  ( ph  ->  G  e.  ( Z H X ) )
Assertion
Ref Expression
yon12  |-  ( ph  ->  ( ( ( Z ( 2nd `  (
( 1st `  Y
) `  X )
) W ) `  F ) `  G
)  =  ( G ( <. W ,  Z >.  .x.  X ) F ) )

Proof of Theorem yon12
StepHypRef Expression
1 yon11.y . . . . . . . . . 10  |-  Y  =  (Yon `  C )
2 yon11.c . . . . . . . . . 10  |-  ( ph  ->  C  e.  Cat )
3 eqid 2443 . . . . . . . . . 10  |-  (oppCat `  C )  =  (oppCat `  C )
4 eqid 2443 . . . . . . . . . 10  |-  (HomF `  (oppCat `  C ) )  =  (HomF
`  (oppCat `  C )
)
51, 2, 3, 4yonval 15404 . . . . . . . . 9  |-  ( ph  ->  Y  =  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) ) )
65fveq2d 5860 . . . . . . . 8  |-  ( ph  ->  ( 1st `  Y
)  =  ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) )
76fveq1d 5858 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  Y
) `  X )  =  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) )
87fveq2d 5860 . . . . . 6  |-  ( ph  ->  ( 2nd `  (
( 1st `  Y
) `  X )
)  =  ( 2nd `  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) ) )
98oveqd 6298 . . . . 5  |-  ( ph  ->  ( Z ( 2nd `  ( ( 1st `  Y
) `  X )
) W )  =  ( Z ( 2nd `  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) ) W ) )
109fveq1d 5858 . . . 4  |-  ( ph  ->  ( ( Z ( 2nd `  ( ( 1st `  Y ) `
 X ) ) W ) `  F
)  =  ( ( Z ( 2nd `  (
( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) ) W ) `  F ) )
11 eqid 2443 . . . . 5  |-  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) )  =  (
<. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) )
12 yon11.b . . . . 5  |-  B  =  ( Base `  C
)
133oppccat 14994 . . . . . 6  |-  ( C  e.  Cat  ->  (oppCat `  C )  e.  Cat )
142, 13syl 16 . . . . 5  |-  ( ph  ->  (oppCat `  C )  e.  Cat )
15 eqid 2443 . . . . . 6  |-  ( SetCat ` 
ran  ( Hom f  `  C ) )  =  ( SetCat ` 
ran  ( Hom f  `  C ) )
16 fvex 5866 . . . . . . . 8  |-  ( Hom f  `  C )  e.  _V
1716rnex 6719 . . . . . . 7  |-  ran  ( Hom f  `  C )  e.  _V
1817a1i 11 . . . . . 6  |-  ( ph  ->  ran  ( Hom f  `  C )  e.  _V )
19 ssid 3508 . . . . . . 7  |-  ran  ( Hom f  `  C )  C_  ran  ( Hom f  `  C )
2019a1i 11 . . . . . 6  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  ran  ( Hom f  `  C
) )
213, 4, 15, 2, 18, 20oppchofcl 15403 . . . . 5  |-  ( ph  ->  (HomF
`  (oppCat `  C )
)  e.  ( ( C  X.c  (oppCat `  C )
)  Func  ( SetCat ` 
ran  ( Hom f  `  C ) ) ) )
223, 12oppcbas 14990 . . . . 5  |-  B  =  ( Base `  (oppCat `  C ) )
23 yon11.p . . . . 5  |-  ( ph  ->  X  e.  B )
24 eqid 2443 . . . . 5  |-  ( ( 1st `  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) ) ) `  X )  =  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X )
25 yon11.z . . . . 5  |-  ( ph  ->  Z  e.  B )
26 eqid 2443 . . . . 5  |-  ( Hom  `  (oppCat `  C )
)  =  ( Hom  `  (oppCat `  C )
)
27 eqid 2443 . . . . 5  |-  ( Id
`  C )  =  ( Id `  C
)
28 yon12.w . . . . 5  |-  ( ph  ->  W  e.  B )
29 yon12.f . . . . . 6  |-  ( ph  ->  F  e.  ( W H Z ) )
30 yon11.h . . . . . . 7  |-  H  =  ( Hom  `  C
)
3130, 3oppchom 14987 . . . . . 6  |-  ( Z ( Hom  `  (oppCat `  C ) ) W )  =  ( W H Z )
3229, 31syl6eleqr 2542 . . . . 5  |-  ( ph  ->  F  e.  ( Z ( Hom  `  (oppCat `  C ) ) W ) )
3311, 12, 2, 14, 21, 22, 23, 24, 25, 26, 27, 28, 32curf12 15370 . . . 4  |-  ( ph  ->  ( ( Z ( 2nd `  ( ( 1st `  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) ) ) `  X ) ) W ) `  F )  =  ( ( ( Id `  C ) `
 X ) (
<. X ,  Z >. ( 2nd `  (HomF `  (oppCat `  C ) ) )
<. X ,  W >. ) F ) )
3410, 33eqtrd 2484 . . 3  |-  ( ph  ->  ( ( Z ( 2nd `  ( ( 1st `  Y ) `
 X ) ) W ) `  F
)  =  ( ( ( Id `  C
) `  X )
( <. X ,  Z >. ( 2nd `  (HomF `  (oppCat `  C ) ) )
<. X ,  W >. ) F ) )
3534fveq1d 5858 . 2  |-  ( ph  ->  ( ( ( Z ( 2nd `  (
( 1st `  Y
) `  X )
) W ) `  F ) `  G
)  =  ( ( ( ( Id `  C ) `  X
) ( <. X ,  Z >. ( 2nd `  (HomF `  (oppCat `  C ) ) )
<. X ,  W >. ) F ) `  G
) )
36 eqid 2443 . . 3  |-  (comp `  (oppCat `  C ) )  =  (comp `  (oppCat `  C ) )
3712, 30, 27, 2, 23catidcl 14956 . . . 4  |-  ( ph  ->  ( ( Id `  C ) `  X
)  e.  ( X H X ) )
3830, 3oppchom 14987 . . . 4  |-  ( X ( Hom  `  (oppCat `  C ) ) X )  =  ( X H X )
3937, 38syl6eleqr 2542 . . 3  |-  ( ph  ->  ( ( Id `  C ) `  X
)  e.  ( X ( Hom  `  (oppCat `  C ) ) X ) )
40 yon12.g . . . 4  |-  ( ph  ->  G  e.  ( Z H X ) )
4130, 3oppchom 14987 . . . 4  |-  ( X ( Hom  `  (oppCat `  C ) ) Z )  =  ( Z H X )
4240, 41syl6eleqr 2542 . . 3  |-  ( ph  ->  G  e.  ( X ( Hom  `  (oppCat `  C ) ) Z ) )
434, 14, 22, 26, 23, 25, 23, 28, 36, 39, 32, 42hof2 15400 . 2  |-  ( ph  ->  ( ( ( ( Id `  C ) `
 X ) (
<. X ,  Z >. ( 2nd `  (HomF `  (oppCat `  C ) ) )
<. X ,  W >. ) F ) `  G
)  =  ( ( F ( <. X ,  Z >. (comp `  (oppCat `  C ) ) W ) G ) (
<. X ,  X >. (comp `  (oppCat `  C )
) W ) ( ( Id `  C
) `  X )
) )
44 yon12.x . . . . 5  |-  .x.  =  (comp `  C )
4512, 44, 3, 23, 25, 28oppcco 14989 . . . 4  |-  ( ph  ->  ( F ( <. X ,  Z >. (comp `  (oppCat `  C )
) W ) G )  =  ( G ( <. W ,  Z >.  .x.  X ) F ) )
4645oveq1d 6296 . . 3  |-  ( ph  ->  ( ( F (
<. X ,  Z >. (comp `  (oppCat `  C )
) W ) G ) ( <. X ,  X >. (comp `  (oppCat `  C ) ) W ) ( ( Id
`  C ) `  X ) )  =  ( ( G (
<. W ,  Z >.  .x. 
X ) F ) ( <. X ,  X >. (comp `  (oppCat `  C
) ) W ) ( ( Id `  C ) `  X
) ) )
4712, 44, 3, 23, 23, 28oppcco 14989 . . 3  |-  ( ph  ->  ( ( G (
<. W ,  Z >.  .x. 
X ) F ) ( <. X ,  X >. (comp `  (oppCat `  C
) ) W ) ( ( Id `  C ) `  X
) )  =  ( ( ( Id `  C ) `  X
) ( <. W ,  X >.  .x.  X )
( G ( <. W ,  Z >.  .x. 
X ) F ) ) )
4812, 30, 44, 2, 28, 25, 23, 29, 40catcocl 14959 . . . 4  |-  ( ph  ->  ( G ( <. W ,  Z >.  .x. 
X ) F )  e.  ( W H X ) )
4912, 30, 27, 2, 28, 44, 23, 48catlid 14957 . . 3  |-  ( ph  ->  ( ( ( Id
`  C ) `  X ) ( <. W ,  X >.  .x. 
X ) ( G ( <. W ,  Z >.  .x.  X ) F ) )  =  ( G ( <. W ,  Z >.  .x.  X ) F ) )
5046, 47, 493eqtrd 2488 . 2  |-  ( ph  ->  ( ( F (
<. X ,  Z >. (comp `  (oppCat `  C )
) W ) G ) ( <. X ,  X >. (comp `  (oppCat `  C ) ) W ) ( ( Id
`  C ) `  X ) )  =  ( G ( <. W ,  Z >.  .x. 
X ) F ) )
5135, 43, 503eqtrd 2488 1  |-  ( ph  ->  ( ( ( Z ( 2nd `  (
( 1st `  Y
) `  X )
) W ) `  F ) `  G
)  =  ( G ( <. W ,  Z >.  .x.  X ) F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804   _Vcvv 3095    C_ wss 3461   <.cop 4020   ran crn 4990   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   Basecbs 14509   Hom chom 14585  compcco 14586   Catccat 14938   Idccid 14939   Hom f chomf 14940  oppCatcoppc 14983   SetCatcsetc 15276   curryF ccurf 15353  HomFchof 15391  Yoncyon 15392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-fz 11682  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-hom 14598  df-cco 14599  df-cat 14942  df-cid 14943  df-homf 14944  df-comf 14945  df-oppc 14984  df-func 15101  df-setc 15277  df-xpc 15315  df-curf 15357  df-hof 15393  df-yon 15394
This theorem is referenced by:  yonedalem4c  15420  yonedalem3b  15422  yonedainv  15424  yonffthlem  15425
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