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Theorem yon11 15382
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 2462 . . . . . . 7  |-  (oppCat `  C )  =  (oppCat `  C )
4 eqid 2462 . . . . . . 7  |-  (HomF `  (oppCat `  C ) )  =  (HomF
`  (oppCat `  C )
)
51, 2, 3, 4yonval 15379 . . . . . 6  |-  ( ph  ->  Y  =  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) ) )
65fveq2d 5863 . . . . 5  |-  ( ph  ->  ( 1st `  Y
)  =  ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) )
76fveq1d 5861 . . . 4  |-  ( ph  ->  ( ( 1st `  Y
) `  X )  =  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) )
87fveq2d 5863 . . 3  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
)  =  ( 1st `  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) ) )
98fveq1d 5861 . 2  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  Z )  =  ( ( 1st `  ( ( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) ) `  Z ) )
10 eqid 2462 . . 3  |-  ( <. C ,  (oppCat `  C
) >. curryF  (HomF `  (oppCat `  C )
) )  =  (
<. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) )
11 yon11.b . . 3  |-  B  =  ( Base `  C
)
123oppccat 14969 . . . 4  |-  ( C  e.  Cat  ->  (oppCat `  C )  e.  Cat )
132, 12syl 16 . . 3  |-  ( ph  ->  (oppCat `  C )  e.  Cat )
14 eqid 2462 . . . 4  |-  ( SetCat ` 
ran  ( Hom f  `  C ) )  =  ( SetCat ` 
ran  ( Hom f  `  C ) )
15 fvex 5869 . . . . . 6  |-  ( Hom f  `  C )  e.  _V
1615rnex 6710 . . . . 5  |-  ran  ( Hom f  `  C )  e.  _V
1716a1i 11 . . . 4  |-  ( ph  ->  ran  ( Hom f  `  C )  e.  _V )
18 ssid 3518 . . . . 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 15378 . . 3  |-  ( ph  ->  (HomF
`  (oppCat `  C )
)  e.  ( ( C  X.c  (oppCat `  C )
)  Func  ( SetCat ` 
ran  ( Hom f  `  C ) ) ) )
213, 11oppcbas 14965 . . 3  |-  B  =  ( Base `  (oppCat `  C ) )
22 yon11.p . . 3  |-  ( ph  ->  X  e.  B )
23 eqid 2462 . . 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 15344 . 2  |-  ( ph  ->  ( ( 1st `  (
( 1st `  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) ) ) `  X ) ) `  Z )  =  ( X ( 1st `  (HomF `  (oppCat `  C ) ) ) Z ) )
26 eqid 2462 . . . 4  |-  ( Hom  `  (oppCat `  C )
)  =  ( Hom  `  (oppCat `  C )
)
274, 13, 21, 26, 22, 24hof1 15372 . . 3  |-  ( ph  ->  ( X ( 1st `  (HomF
`  (oppCat `  C )
) ) Z )  =  ( X ( Hom  `  (oppCat `  C
) ) Z ) )
28 yon11.h . . . 4  |-  H  =  ( Hom  `  C
)
2928, 3oppchom 14962 . . 3  |-  ( X ( Hom  `  (oppCat `  C ) ) Z )  =  ( Z H X )
3027, 29syl6eq 2519 . 2  |-  ( ph  ->  ( X ( 1st `  (HomF
`  (oppCat `  C )
) ) Z )  =  ( Z H X ) )
319, 25, 303eqtrd 2507 1  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  Z )  =  ( Z H X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   _Vcvv 3108    C_ wss 3471   <.cop 4028   ran crn 4995   ` cfv 5581  (class class class)co 6277   1stc1st 6774   Basecbs 14481   Hom chom 14557   Catccat 14910   Hom f chomf 14912  oppCatcoppc 14958   SetCatcsetc 15251   curryF ccurf 15328  HomFchof 15366  Yoncyon 15367
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 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
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 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-fz 11664  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-hom 14570  df-cco 14571  df-cat 14914  df-cid 14915  df-homf 14916  df-comf 14917  df-oppc 14959  df-func 15076  df-setc 15252  df-xpc 15290  df-curf 15332  df-hof 15368  df-yon 15369
This theorem is referenced by:  yonedalem3a  15392  yonedalem4c  15395  yonedalem3b  15397  yonedainv  15399  yonffthlem  15400  yoniso  15403
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