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Theorem yon12 15179
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 2451 . . . . . . . . . 10  |-  (oppCat `  C )  =  (oppCat `  C )
4 eqid 2451 . . . . . . . . . 10  |-  (HomF `  (oppCat `  C ) )  =  (HomF
`  (oppCat `  C )
)
51, 2, 3, 4yonval 15175 . . . . . . . . 9  |-  ( ph  ->  Y  =  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) ) )
65fveq2d 5795 . . . . . . . 8  |-  ( ph  ->  ( 1st `  Y
)  =  ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) )
76fveq1d 5793 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  Y
) `  X )  =  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) )
87fveq2d 5795 . . . . . 6  |-  ( ph  ->  ( 2nd `  (
( 1st `  Y
) `  X )
)  =  ( 2nd `  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) ) )
98oveqd 6209 . . . . 5  |-  ( ph  ->  ( Z ( 2nd `  ( ( 1st `  Y
) `  X )
) W )  =  ( Z ( 2nd `  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) ) W ) )
109fveq1d 5793 . . . 4  |-  ( ph  ->  ( ( Z ( 2nd `  ( ( 1st `  Y ) `
 X ) ) W ) `  F
)  =  ( ( Z ( 2nd `  (
( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) ) W ) `  F ) )
11 eqid 2451 . . . . 5  |-  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) )  =  (
<. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) )
12 yon11.b . . . . 5  |-  B  =  ( Base `  C
)
133oppccat 14765 . . . . . 6  |-  ( C  e.  Cat  ->  (oppCat `  C )  e.  Cat )
142, 13syl 16 . . . . 5  |-  ( ph  ->  (oppCat `  C )  e.  Cat )
15 eqid 2451 . . . . . 6  |-  ( SetCat ` 
ran  ( Hom f  `  C ) )  =  ( SetCat ` 
ran  ( Hom f  `  C ) )
16 fvex 5801 . . . . . . . 8  |-  ( Hom f  `  C )  e.  _V
1716rnex 6614 . . . . . . 7  |-  ran  ( Hom f  `  C )  e.  _V
1817a1i 11 . . . . . 6  |-  ( ph  ->  ran  ( Hom f  `  C )  e.  _V )
19 ssid 3475 . . . . . . 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 15174 . . . . 5  |-  ( ph  ->  (HomF
`  (oppCat `  C )
)  e.  ( ( C  X.c  (oppCat `  C )
)  Func  ( SetCat ` 
ran  ( Hom f  `  C ) ) ) )
223, 12oppcbas 14761 . . . . 5  |-  B  =  ( Base `  (oppCat `  C ) )
23 yon11.p . . . . 5  |-  ( ph  ->  X  e.  B )
24 eqid 2451 . . . . 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 2451 . . . . 5  |-  ( Hom  `  (oppCat `  C )
)  =  ( Hom  `  (oppCat `  C )
)
27 eqid 2451 . . . . 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 14758 . . . . . 6  |-  ( Z ( Hom  `  (oppCat `  C ) ) W )  =  ( W H Z )
3229, 31syl6eleqr 2550 . . . . 5  |-  ( ph  ->  F  e.  ( Z ( Hom  `  (oppCat `  C ) ) W ) )
3311, 12, 2, 14, 21, 22, 23, 24, 25, 26, 27, 28, 32curf12 15141 . . . 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 2492 . . 3  |-  ( ph  ->  ( ( Z ( 2nd `  ( ( 1st `  Y ) `
 X ) ) W ) `  F
)  =  ( ( ( Id `  C
) `  X )
( <. X ,  Z >. ( 2nd `  (HomF `  (oppCat `  C ) ) )
<. X ,  W >. ) F ) )
3534fveq1d 5793 . 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 2451 . . 3  |-  (comp `  (oppCat `  C ) )  =  (comp `  (oppCat `  C ) )
3712, 30, 27, 2, 23catidcl 14724 . . . 4  |-  ( ph  ->  ( ( Id `  C ) `  X
)  e.  ( X H X ) )
3830, 3oppchom 14758 . . . 4  |-  ( X ( Hom  `  (oppCat `  C ) ) X )  =  ( X H X )
3937, 38syl6eleqr 2550 . . 3  |-  ( ph  ->  ( ( Id `  C ) `  X
)  e.  ( X ( Hom  `  (oppCat `  C ) ) X ) )
40 yon12.g . . . 4  |-  ( ph  ->  G  e.  ( Z H X ) )
4130, 3oppchom 14758 . . . 4  |-  ( X ( Hom  `  (oppCat `  C ) ) Z )  =  ( Z H X )
4240, 41syl6eleqr 2550 . . 3  |-  ( ph  ->  G  e.  ( X ( Hom  `  (oppCat `  C ) ) Z ) )
434, 14, 22, 26, 23, 25, 23, 28, 36, 39, 32, 42hof2 15171 . 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 14760 . . . 4  |-  ( ph  ->  ( F ( <. X ,  Z >. (comp `  (oppCat `  C )
) W ) G )  =  ( G ( <. W ,  Z >.  .x.  X ) F ) )
4645oveq1d 6207 . . 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 14760 . . 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 14727 . . . 4  |-  ( ph  ->  ( G ( <. W ,  Z >.  .x. 
X ) F )  e.  ( W H X ) )
4912, 30, 27, 2, 28, 44, 23, 48catlid 14725 . . 3  |-  ( ph  ->  ( ( ( Id
`  C ) `  X ) ( <. W ,  X >.  .x. 
X ) ( G ( <. W ,  Z >.  .x.  X ) F ) )  =  ( G ( <. W ,  Z >.  .x.  X ) F ) )
5046, 47, 493eqtrd 2496 . 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 2496 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 1370    e. wcel 1758   _Vcvv 3070    C_ wss 3428   <.cop 3983   ran crn 4941   ` cfv 5518  (class class class)co 6192   1stc1st 6677   2ndc2nd 6678   Basecbs 14278   Hom chom 14353  compcco 14354   Catccat 14706   Idccid 14707   Hom f chomf 14708  oppCatcoppc 14754   SetCatcsetc 15047   curryF ccurf 15124  HomFchof 15162  Yoncyon 15163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-tpos 6847  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-ixp 7366  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-10 10491  df-n0 10683  df-z 10750  df-dec 10859  df-uz 10965  df-fz 11541  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-hom 14366  df-cco 14367  df-cat 14710  df-cid 14711  df-homf 14712  df-comf 14713  df-oppc 14755  df-func 14872  df-setc 15048  df-xpc 15086  df-curf 15128  df-hof 15164  df-yon 15165
This theorem is referenced by:  yonedalem4c  15191  yonedalem3b  15193  yonedainv  15195  yonffthlem  15196
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