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Theorem yon2 16095
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 2420 . . . . . . . . 9  |-  (oppCat `  C )  =  (oppCat `  C )
4 eqid 2420 . . . . . . . . 9  |-  (HomF `  (oppCat `  C ) )  =  (HomF
`  (oppCat `  C )
)
51, 2, 3, 4yonval 16090 . . . . . . . 8  |-  ( ph  ->  Y  =  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) ) )
65fveq2d 5876 . . . . . . 7  |-  ( ph  ->  ( 2nd `  Y
)  =  ( 2nd `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) )
76oveqd 6313 . . . . . 6  |-  ( ph  ->  ( X ( 2nd `  Y ) Z )  =  ( X ( 2nd `  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) ) ) Z ) )
87fveq1d 5874 . . . . 5  |-  ( ph  ->  ( ( X ( 2nd `  Y ) Z ) `  F
)  =  ( ( X ( 2nd `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) Z ) `  F ) )
98fveq1d 5874 . . . 4  |-  ( ph  ->  ( ( ( X ( 2nd `  Y
) Z ) `  F ) `  W
)  =  ( ( ( X ( 2nd `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) Z ) `  F ) `
 W ) )
10 eqid 2420 . . . . 5  |-  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) )  =  (
<. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) )
11 yon11.b . . . . 5  |-  B  =  ( Base `  C
)
123oppccat 15571 . . . . . 6  |-  ( C  e.  Cat  ->  (oppCat `  C )  e.  Cat )
132, 12syl 17 . . . . 5  |-  ( ph  ->  (oppCat `  C )  e.  Cat )
14 eqid 2420 . . . . . 6  |-  ( SetCat ` 
ran  ( Hom f  `  C ) )  =  ( SetCat ` 
ran  ( Hom f  `  C ) )
15 fvex 5882 . . . . . . . 8  |-  ( Hom f  `  C )  e.  _V
1615rnex 6732 . . . . . . 7  |-  ran  ( Hom f  `  C )  e.  _V
1716a1i 11 . . . . . 6  |-  ( ph  ->  ran  ( Hom f  `  C )  e.  _V )
18 ssid 3480 . . . . . . 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 16089 . . . . 5  |-  ( ph  ->  (HomF
`  (oppCat `  C )
)  e.  ( ( C  X.c  (oppCat `  C )
)  Func  ( SetCat ` 
ran  ( Hom f  `  C ) ) ) )
213, 11oppcbas 15567 . . . . 5  |-  B  =  ( Base `  (oppCat `  C ) )
22 yon11.h . . . . 5  |-  H  =  ( Hom  `  C
)
23 eqid 2420 . . . . 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 2420 . . . . 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 16059 . . . 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 2461 . . 3  |-  ( ph  ->  ( ( ( X ( 2nd `  Y
) Z ) `  F ) `  W
)  =  ( F ( <. X ,  W >. ( 2nd `  (HomF `  (oppCat `  C ) ) )
<. Z ,  W >. ) ( ( Id `  (oppCat `  C ) ) `
 W ) ) )
3130fveq1d 5874 . 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 2420 . . 3  |-  ( Hom  `  (oppCat `  C )
)  =  ( Hom  `  (oppCat `  C )
)
33 eqid 2420 . . 3  |-  (comp `  (oppCat `  C ) )  =  (comp `  (oppCat `  C ) )
3422, 3oppchom 15564 . . . 4  |-  ( Z ( Hom  `  (oppCat `  C ) ) X )  =  ( X H Z )
3526, 34syl6eleqr 2519 . . 3  |-  ( ph  ->  F  e.  ( Z ( Hom  `  (oppCat `  C ) ) X ) )
3621, 32, 23, 13, 28catidcl 15532 . . 3  |-  ( ph  ->  ( ( Id `  (oppCat `  C ) ) `
 W )  e.  ( W ( Hom  `  (oppCat `  C )
) W ) )
37 yon2.g . . . 4  |-  ( ph  ->  G  e.  ( W H X ) )
3822, 3oppchom 15564 . . . 4  |-  ( X ( Hom  `  (oppCat `  C ) ) W )  =  ( W H X )
3937, 38syl6eleqr 2519 . . 3  |-  ( ph  ->  G  e.  ( X ( Hom  `  (oppCat `  C ) ) W ) )
404, 13, 21, 32, 24, 28, 25, 28, 33, 35, 36, 39hof2 16086 . 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 15533 . . . 4  |-  ( ph  ->  ( ( ( Id
`  (oppCat `  C )
) `  W )
( <. X ,  W >. (comp `  (oppCat `  C
) ) W ) G )  =  G )
4241oveq1d 6311 . . 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 15566 . . 3  |-  ( ph  ->  ( G ( <. Z ,  X >. (comp `  (oppCat `  C )
) W ) F )  =  ( F ( <. W ,  X >.  .x.  Z ) G ) )
4542, 44eqtrd 2461 . 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 2465 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 1437    e. wcel 1867   _Vcvv 3078    C_ wss 3433   <.cop 3999   ran crn 4846   ` cfv 5592  (class class class)co 6296   2ndc2nd 6797   Basecbs 15073   Hom chom 15153  compcco 15154   Catccat 15514   Idccid 15515   Hom f chomf 15516  oppCatcoppc 15560   SetCatcsetc 15914   curryF ccurf 16039  HomFchof 16077  Yoncyon 16078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-tpos 6972  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-fz 11772  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-hom 15166  df-cco 15167  df-cat 15518  df-cid 15519  df-homf 15520  df-comf 15521  df-oppc 15561  df-func 15707  df-setc 15915  df-xpc 16001  df-curf 16043  df-hof 16079  df-yon 16080
This theorem is referenced by:  yonedalem3b  16108  yonffthlem  16111
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