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Theorem yon12 15067
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 2438 . . . . . . . . . 10  |-  (oppCat `  C )  =  (oppCat `  C )
4 eqid 2438 . . . . . . . . . 10  |-  (HomF `  (oppCat `  C ) )  =  (HomF
`  (oppCat `  C )
)
51, 2, 3, 4yonval 15063 . . . . . . . . 9  |-  ( ph  ->  Y  =  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) ) )
65fveq2d 5690 . . . . . . . 8  |-  ( ph  ->  ( 1st `  Y
)  =  ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) )
76fveq1d 5688 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  Y
) `  X )  =  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) )
87fveq2d 5690 . . . . . 6  |-  ( ph  ->  ( 2nd `  (
( 1st `  Y
) `  X )
)  =  ( 2nd `  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) ) )
98oveqd 6103 . . . . 5  |-  ( ph  ->  ( Z ( 2nd `  ( ( 1st `  Y
) `  X )
) W )  =  ( Z ( 2nd `  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) ) W ) )
109fveq1d 5688 . . . 4  |-  ( ph  ->  ( ( Z ( 2nd `  ( ( 1st `  Y ) `
 X ) ) W ) `  F
)  =  ( ( Z ( 2nd `  (
( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) ) W ) `  F ) )
11 eqid 2438 . . . . 5  |-  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) )  =  (
<. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) )
12 yon11.b . . . . 5  |-  B  =  ( Base `  C
)
133oppccat 14653 . . . . . 6  |-  ( C  e.  Cat  ->  (oppCat `  C )  e.  Cat )
142, 13syl 16 . . . . 5  |-  ( ph  ->  (oppCat `  C )  e.  Cat )
15 eqid 2438 . . . . . 6  |-  ( SetCat ` 
ran  ( Hom f  `  C ) )  =  ( SetCat ` 
ran  ( Hom f  `  C ) )
16 fvex 5696 . . . . . . . 8  |-  ( Hom f  `  C )  e.  _V
1716rnex 6507 . . . . . . 7  |-  ran  ( Hom f  `  C )  e.  _V
1817a1i 11 . . . . . 6  |-  ( ph  ->  ran  ( Hom f  `  C )  e.  _V )
19 ssid 3370 . . . . . . 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 15062 . . . . 5  |-  ( ph  ->  (HomF
`  (oppCat `  C )
)  e.  ( ( C  X.c  (oppCat `  C )
)  Func  ( SetCat ` 
ran  ( Hom f  `  C ) ) ) )
223, 12oppcbas 14649 . . . . 5  |-  B  =  ( Base `  (oppCat `  C ) )
23 yon11.p . . . . 5  |-  ( ph  ->  X  e.  B )
24 eqid 2438 . . . . 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 2438 . . . . 5  |-  ( Hom  `  (oppCat `  C )
)  =  ( Hom  `  (oppCat `  C )
)
27 eqid 2438 . . . . 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 14646 . . . . . 6  |-  ( Z ( Hom  `  (oppCat `  C ) ) W )  =  ( W H Z )
3229, 31syl6eleqr 2529 . . . . 5  |-  ( ph  ->  F  e.  ( Z ( Hom  `  (oppCat `  C ) ) W ) )
3311, 12, 2, 14, 21, 22, 23, 24, 25, 26, 27, 28, 32curf12 15029 . . . 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 2470 . . 3  |-  ( ph  ->  ( ( Z ( 2nd `  ( ( 1st `  Y ) `
 X ) ) W ) `  F
)  =  ( ( ( Id `  C
) `  X )
( <. X ,  Z >. ( 2nd `  (HomF `  (oppCat `  C ) ) )
<. X ,  W >. ) F ) )
3534fveq1d 5688 . 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 2438 . . 3  |-  (comp `  (oppCat `  C ) )  =  (comp `  (oppCat `  C ) )
3712, 30, 27, 2, 23catidcl 14612 . . . 4  |-  ( ph  ->  ( ( Id `  C ) `  X
)  e.  ( X H X ) )
3830, 3oppchom 14646 . . . 4  |-  ( X ( Hom  `  (oppCat `  C ) ) X )  =  ( X H X )
3937, 38syl6eleqr 2529 . . 3  |-  ( ph  ->  ( ( Id `  C ) `  X
)  e.  ( X ( Hom  `  (oppCat `  C ) ) X ) )
40 yon12.g . . . 4  |-  ( ph  ->  G  e.  ( Z H X ) )
4130, 3oppchom 14646 . . . 4  |-  ( X ( Hom  `  (oppCat `  C ) ) Z )  =  ( Z H X )
4240, 41syl6eleqr 2529 . . 3  |-  ( ph  ->  G  e.  ( X ( Hom  `  (oppCat `  C ) ) Z ) )
434, 14, 22, 26, 23, 25, 23, 28, 36, 39, 32, 42hof2 15059 . 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 14648 . . . 4  |-  ( ph  ->  ( F ( <. X ,  Z >. (comp `  (oppCat `  C )
) W ) G )  =  ( G ( <. W ,  Z >.  .x.  X ) F ) )
4645oveq1d 6101 . . 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 14648 . . 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 14615 . . . 4  |-  ( ph  ->  ( G ( <. W ,  Z >.  .x. 
X ) F )  e.  ( W H X ) )
4912, 30, 27, 2, 28, 44, 23, 48catlid 14613 . . 3  |-  ( ph  ->  ( ( ( Id
`  C ) `  X ) ( <. W ,  X >.  .x. 
X ) ( G ( <. W ,  Z >.  .x.  X ) F ) )  =  ( G ( <. W ,  Z >.  .x.  X ) F ) )
5046, 47, 493eqtrd 2474 . 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 2474 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 1369    e. wcel 1756   _Vcvv 2967    C_ wss 3323   <.cop 3878   ran crn 4836   ` cfv 5413  (class class class)co 6086   1stc1st 6570   2ndc2nd 6571   Basecbs 14166   Hom chom 14241  compcco 14242   Catccat 14594   Idccid 14595   Hom f chomf 14596  oppCatcoppc 14642   SetCatcsetc 14935   curryF ccurf 15012  HomFchof 15050  Yoncyon 15051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-tpos 6740  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-hom 14254  df-cco 14255  df-cat 14598  df-cid 14599  df-homf 14600  df-comf 14601  df-oppc 14643  df-func 14760  df-setc 14936  df-xpc 14974  df-curf 15016  df-hof 15052  df-yon 15053
This theorem is referenced by:  yonedalem4c  15079  yonedalem3b  15081  yonedainv  15083  yonffthlem  15084
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