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Theorem yonpropd 15861
Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same Yoneda functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
hofpropd.1  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
hofpropd.2  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
hofpropd.c  |-  ( ph  ->  C  e.  Cat )
hofpropd.d  |-  ( ph  ->  D  e.  Cat )
Assertion
Ref Expression
yonpropd  |-  ( ph  ->  (Yon `  C )  =  (Yon `  D )
)

Proof of Theorem yonpropd
StepHypRef Expression
1 hofpropd.1 . . . 4  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  D ) )
2 hofpropd.2 . . . 4  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
31oppchomfpropd 15339 . . . 4  |-  ( ph  ->  ( Hom f  `  (oppCat `  C
) )  =  ( Hom f  `  (oppCat `  D )
) )
41, 2oppccomfpropd 15340 . . . 4  |-  ( ph  ->  (compf `  (oppCat `  C )
)  =  (compf `  (oppCat `  D ) ) )
5 hofpropd.c . . . 4  |-  ( ph  ->  C  e.  Cat )
6 hofpropd.d . . . 4  |-  ( ph  ->  D  e.  Cat )
7 eqid 2402 . . . . . 6  |-  (oppCat `  C )  =  (oppCat `  C )
87oppccat 15335 . . . . 5  |-  ( C  e.  Cat  ->  (oppCat `  C )  e.  Cat )
95, 8syl 17 . . . 4  |-  ( ph  ->  (oppCat `  C )  e.  Cat )
10 eqid 2402 . . . . . 6  |-  (oppCat `  D )  =  (oppCat `  D )
1110oppccat 15335 . . . . 5  |-  ( D  e.  Cat  ->  (oppCat `  D )  e.  Cat )
126, 11syl 17 . . . 4  |-  ( ph  ->  (oppCat `  D )  e.  Cat )
13 eqid 2402 . . . . 5  |-  (HomF `  (oppCat `  C ) )  =  (HomF
`  (oppCat `  C )
)
14 eqid 2402 . . . . 5  |-  ( SetCat ` 
ran  ( Hom f  `  C ) )  =  ( SetCat ` 
ran  ( Hom f  `  C ) )
15 fvex 5859 . . . . . . 7  |-  ( Hom f  `  C )  e.  _V
1615rnex 6718 . . . . . 6  |-  ran  ( Hom f  `  C )  e.  _V
1716a1i 11 . . . . 5  |-  ( ph  ->  ran  ( Hom f  `  C )  e.  _V )
18 ssid 3461 . . . . . 6  |-  ran  ( Hom f  `  C )  C_  ran  ( Hom f  `  C )
1918a1i 11 . . . . 5  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  ran  ( Hom f  `  C
) )
207, 13, 14, 5, 17, 19oppchofcl 15853 . . . 4  |-  ( ph  ->  (HomF
`  (oppCat `  C )
)  e.  ( ( C  X.c  (oppCat `  C )
)  Func  ( SetCat ` 
ran  ( Hom f  `  C ) ) ) )
211, 2, 3, 4, 5, 6, 9, 12, 20curfpropd 15826 . . 3  |-  ( ph  ->  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) )  =  (
<. D ,  (oppCat `  D ) >. curryF  (HomF
`  (oppCat `  C )
) ) )
223, 4, 9, 12hofpropd 15860 . . . 4  |-  ( ph  ->  (HomF
`  (oppCat `  C )
)  =  (HomF `  (oppCat `  D ) ) )
2322oveq2d 6294 . . 3  |-  ( ph  ->  ( <. D ,  (oppCat `  D ) >. curryF  (HomF
`  (oppCat `  C )
) )  =  (
<. D ,  (oppCat `  D ) >. curryF  (HomF
`  (oppCat `  D )
) ) )
2421, 23eqtrd 2443 . 2  |-  ( ph  ->  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) )  =  (
<. D ,  (oppCat `  D ) >. curryF  (HomF
`  (oppCat `  D )
) ) )
25 eqid 2402 . . 3  |-  (Yon `  C )  =  (Yon
`  C )
2625, 5, 7, 13yonval 15854 . 2  |-  ( ph  ->  (Yon `  C )  =  ( <. C , 
(oppCat `  C ) >. curryF  (HomF `  (oppCat `  C ) ) ) )
27 eqid 2402 . . 3  |-  (Yon `  D )  =  (Yon
`  D )
28 eqid 2402 . . 3  |-  (HomF `  (oppCat `  D ) )  =  (HomF
`  (oppCat `  D )
)
2927, 6, 10, 28yonval 15854 . 2  |-  ( ph  ->  (Yon `  D )  =  ( <. D , 
(oppCat `  D ) >. curryF  (HomF `  (oppCat `  D ) ) ) )
3024, 26, 293eqtr4d 2453 1  |-  ( ph  ->  (Yon `  C )  =  (Yon `  D )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   _Vcvv 3059    C_ wss 3414   <.cop 3978   ran crn 4824   ` cfv 5569  (class class class)co 6278   Catccat 15278   Hom f chomf 15280  compfccomf 15281  oppCatcoppc 15324   SetCatcsetc 15678   curryF ccurf 15803  HomFchof 15841  Yoncyon 15842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-tpos 6958  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-fz 11727  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-hom 14933  df-cco 14934  df-cat 15282  df-cid 15283  df-homf 15284  df-comf 15285  df-oppc 15325  df-func 15471  df-setc 15679  df-xpc 15765  df-curf 15807  df-hof 15843  df-yon 15844
This theorem is referenced by: (None)
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