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Theorem yon12 15381
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 2460 . . . . . . . . . 10  |-  (oppCat `  C )  =  (oppCat `  C )
4 eqid 2460 . . . . . . . . . 10  |-  (HomF `  (oppCat `  C ) )  =  (HomF
`  (oppCat `  C )
)
51, 2, 3, 4yonval 15377 . . . . . . . . 9  |-  ( ph  ->  Y  =  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) ) )
65fveq2d 5861 . . . . . . . 8  |-  ( ph  ->  ( 1st `  Y
)  =  ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) )
76fveq1d 5859 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  Y
) `  X )  =  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) )
87fveq2d 5861 . . . . . 6  |-  ( ph  ->  ( 2nd `  (
( 1st `  Y
) `  X )
)  =  ( 2nd `  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) ) )
98oveqd 6292 . . . . 5  |-  ( ph  ->  ( Z ( 2nd `  ( ( 1st `  Y
) `  X )
) W )  =  ( Z ( 2nd `  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) ) W ) )
109fveq1d 5859 . . . 4  |-  ( ph  ->  ( ( Z ( 2nd `  ( ( 1st `  Y ) `
 X ) ) W ) `  F
)  =  ( ( Z ( 2nd `  (
( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) ) W ) `  F ) )
11 eqid 2460 . . . . 5  |-  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) )  =  (
<. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) )
12 yon11.b . . . . 5  |-  B  =  ( Base `  C
)
133oppccat 14967 . . . . . 6  |-  ( C  e.  Cat  ->  (oppCat `  C )  e.  Cat )
142, 13syl 16 . . . . 5  |-  ( ph  ->  (oppCat `  C )  e.  Cat )
15 eqid 2460 . . . . . 6  |-  ( SetCat ` 
ran  ( Hom f  `  C ) )  =  ( SetCat ` 
ran  ( Hom f  `  C ) )
16 fvex 5867 . . . . . . . 8  |-  ( Hom f  `  C )  e.  _V
1716rnex 6708 . . . . . . 7  |-  ran  ( Hom f  `  C )  e.  _V
1817a1i 11 . . . . . 6  |-  ( ph  ->  ran  ( Hom f  `  C )  e.  _V )
19 ssid 3516 . . . . . . 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 15376 . . . . 5  |-  ( ph  ->  (HomF
`  (oppCat `  C )
)  e.  ( ( C  X.c  (oppCat `  C )
)  Func  ( SetCat ` 
ran  ( Hom f  `  C ) ) ) )
223, 12oppcbas 14963 . . . . 5  |-  B  =  ( Base `  (oppCat `  C ) )
23 yon11.p . . . . 5  |-  ( ph  ->  X  e.  B )
24 eqid 2460 . . . . 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 2460 . . . . 5  |-  ( Hom  `  (oppCat `  C )
)  =  ( Hom  `  (oppCat `  C )
)
27 eqid 2460 . . . . 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 14960 . . . . . 6  |-  ( Z ( Hom  `  (oppCat `  C ) ) W )  =  ( W H Z )
3229, 31syl6eleqr 2559 . . . . 5  |-  ( ph  ->  F  e.  ( Z ( Hom  `  (oppCat `  C ) ) W ) )
3311, 12, 2, 14, 21, 22, 23, 24, 25, 26, 27, 28, 32curf12 15343 . . . 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 2501 . . 3  |-  ( ph  ->  ( ( Z ( 2nd `  ( ( 1st `  Y ) `
 X ) ) W ) `  F
)  =  ( ( ( Id `  C
) `  X )
( <. X ,  Z >. ( 2nd `  (HomF `  (oppCat `  C ) ) )
<. X ,  W >. ) F ) )
3534fveq1d 5859 . 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 2460 . . 3  |-  (comp `  (oppCat `  C ) )  =  (comp `  (oppCat `  C ) )
3712, 30, 27, 2, 23catidcl 14926 . . . 4  |-  ( ph  ->  ( ( Id `  C ) `  X
)  e.  ( X H X ) )
3830, 3oppchom 14960 . . . 4  |-  ( X ( Hom  `  (oppCat `  C ) ) X )  =  ( X H X )
3937, 38syl6eleqr 2559 . . 3  |-  ( ph  ->  ( ( Id `  C ) `  X
)  e.  ( X ( Hom  `  (oppCat `  C ) ) X ) )
40 yon12.g . . . 4  |-  ( ph  ->  G  e.  ( Z H X ) )
4130, 3oppchom 14960 . . . 4  |-  ( X ( Hom  `  (oppCat `  C ) ) Z )  =  ( Z H X )
4240, 41syl6eleqr 2559 . . 3  |-  ( ph  ->  G  e.  ( X ( Hom  `  (oppCat `  C ) ) Z ) )
434, 14, 22, 26, 23, 25, 23, 28, 36, 39, 32, 42hof2 15373 . 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 14962 . . . 4  |-  ( ph  ->  ( F ( <. X ,  Z >. (comp `  (oppCat `  C )
) W ) G )  =  ( G ( <. W ,  Z >.  .x.  X ) F ) )
4645oveq1d 6290 . . 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 14962 . . 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 14929 . . . 4  |-  ( ph  ->  ( G ( <. W ,  Z >.  .x. 
X ) F )  e.  ( W H X ) )
4912, 30, 27, 2, 28, 44, 23, 48catlid 14927 . . 3  |-  ( ph  ->  ( ( ( Id
`  C ) `  X ) ( <. W ,  X >.  .x. 
X ) ( G ( <. W ,  Z >.  .x.  X ) F ) )  =  ( G ( <. W ,  Z >.  .x.  X ) F ) )
5046, 47, 493eqtrd 2505 . 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 2505 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 1374    e. wcel 1762   _Vcvv 3106    C_ wss 3469   <.cop 4026   ran crn 4993   ` cfv 5579  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   Basecbs 14479   Hom chom 14555  compcco 14556   Catccat 14908   Idccid 14909   Hom f chomf 14910  oppCatcoppc 14956   SetCatcsetc 15249   curryF ccurf 15326  HomFchof 15364  Yoncyon 15365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-tpos 6945  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-hom 14568  df-cco 14569  df-cat 14912  df-cid 14913  df-homf 14914  df-comf 14915  df-oppc 14957  df-func 15074  df-setc 15250  df-xpc 15288  df-curf 15330  df-hof 15366  df-yon 15367
This theorem is referenced by:  yonedalem4c  15393  yonedalem3b  15395  yonedainv  15397  yonffthlem  15398
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