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Theorem yon11 15735
Description: Value of the Yoneda embedding at an object. 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 )
Assertion
Ref Expression
yon11  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  Z )  =  ( Z H X ) )

Proof of Theorem yon11
StepHypRef Expression
1 yon11.y . . . . . . 7  |-  Y  =  (Yon `  C )
2 yon11.c . . . . . . 7  |-  ( ph  ->  C  e.  Cat )
3 eqid 2454 . . . . . . 7  |-  (oppCat `  C )  =  (oppCat `  C )
4 eqid 2454 . . . . . . 7  |-  (HomF `  (oppCat `  C ) )  =  (HomF
`  (oppCat `  C )
)
51, 2, 3, 4yonval 15732 . . . . . 6  |-  ( ph  ->  Y  =  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) ) )
65fveq2d 5852 . . . . 5  |-  ( ph  ->  ( 1st `  Y
)  =  ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) )
76fveq1d 5850 . . . 4  |-  ( ph  ->  ( ( 1st `  Y
) `  X )  =  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) )
87fveq2d 5852 . . 3  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
)  =  ( 1st `  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) ) )
98fveq1d 5850 . 2  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  Z )  =  ( ( 1st `  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) ) `  Z ) )
10 eqid 2454 . . 3  |-  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) )  =  (
<. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) )
11 yon11.b . . 3  |-  B  =  ( Base `  C
)
123oppccat 15213 . . . 4  |-  ( C  e.  Cat  ->  (oppCat `  C )  e.  Cat )
132, 12syl 16 . . 3  |-  ( ph  ->  (oppCat `  C )  e.  Cat )
14 eqid 2454 . . . 4  |-  ( SetCat ` 
ran  ( Hom f  `  C ) )  =  ( SetCat ` 
ran  ( Hom f  `  C ) )
15 fvex 5858 . . . . . 6  |-  ( Hom f  `  C )  e.  _V
1615rnex 6707 . . . . 5  |-  ran  ( Hom f  `  C )  e.  _V
1716a1i 11 . . . 4  |-  ( ph  ->  ran  ( Hom f  `  C )  e.  _V )
18 ssid 3508 . . . . 5  |-  ran  ( Hom f  `  C )  C_  ran  ( Hom f  `  C )
1918a1i 11 . . . 4  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  ran  ( Hom f  `  C
) )
203, 4, 14, 2, 17, 19oppchofcl 15731 . . 3  |-  ( ph  ->  (HomF
`  (oppCat `  C )
)  e.  ( ( C  X.c  (oppCat `  C )
)  Func  ( SetCat ` 
ran  ( Hom f  `  C ) ) ) )
213, 11oppcbas 15209 . . 3  |-  B  =  ( Base `  (oppCat `  C ) )
22 yon11.p . . 3  |-  ( ph  ->  X  e.  B )
23 eqid 2454 . . 3  |-  ( ( 1st `  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) ) ) `  X )  =  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X )
24 yon11.z . . 3  |-  ( ph  ->  Z  e.  B )
2510, 11, 2, 13, 20, 21, 22, 23, 24curf11 15697 . 2  |-  ( ph  ->  ( ( 1st `  (
( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) ) `  Z )  =  ( X ( 1st `  (HomF `  (oppCat `  C ) ) ) Z ) )
26 eqid 2454 . . . 4  |-  ( Hom  `  (oppCat `  C )
)  =  ( Hom  `  (oppCat `  C )
)
274, 13, 21, 26, 22, 24hof1 15725 . . 3  |-  ( ph  ->  ( X ( 1st `  (HomF
`  (oppCat `  C )
) ) Z )  =  ( X ( Hom  `  (oppCat `  C
) ) Z ) )
28 yon11.h . . . 4  |-  H  =  ( Hom  `  C
)
2928, 3oppchom 15206 . . 3  |-  ( X ( Hom  `  (oppCat `  C ) ) Z )  =  ( Z H X )
3027, 29syl6eq 2511 . 2  |-  ( ph  ->  ( X ( 1st `  (HomF
`  (oppCat `  C )
) ) Z )  =  ( Z H X ) )
319, 25, 303eqtrd 2499 1  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  Z )  =  ( Z H X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461   <.cop 4022   ran crn 4989   ` cfv 5570  (class class class)co 6270   1stc1st 6771   Basecbs 14719   Hom chom 14798   Catccat 15156   Hom f chomf 15158  oppCatcoppc 15202   SetCatcsetc 15556   curryF ccurf 15681  HomFchof 15719  Yoncyon 15720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  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 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-hom 14811  df-cco 14812  df-cat 15160  df-cid 15161  df-homf 15162  df-comf 15163  df-oppc 15203  df-func 15349  df-setc 15557  df-xpc 15643  df-curf 15685  df-hof 15721  df-yon 15722
This theorem is referenced by:  yonedalem3a  15745  yonedalem4c  15748  yonedalem3b  15750  yonedainv  15752  yonffthlem  15753  yoniso  15756
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