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Theorem yonpropd 15201
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 14788 . . . 4  |-  ( ph  ->  ( Hom f  `  (oppCat `  C
) )  =  ( Hom f  `  (oppCat `  D )
) )
41, 2oppccomfpropd 14789 . . . 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 2454 . . . . . 6  |-  (oppCat `  C )  =  (oppCat `  C )
87oppccat 14784 . . . . 5  |-  ( C  e.  Cat  ->  (oppCat `  C )  e.  Cat )
95, 8syl 16 . . . 4  |-  ( ph  ->  (oppCat `  C )  e.  Cat )
10 eqid 2454 . . . . . 6  |-  (oppCat `  D )  =  (oppCat `  D )
1110oppccat 14784 . . . . 5  |-  ( D  e.  Cat  ->  (oppCat `  D )  e.  Cat )
126, 11syl 16 . . . 4  |-  ( ph  ->  (oppCat `  D )  e.  Cat )
13 eqid 2454 . . . . 5  |-  (HomF `  (oppCat `  C ) )  =  (HomF
`  (oppCat `  C )
)
14 eqid 2454 . . . . 5  |-  ( SetCat ` 
ran  ( Hom f  `  C ) )  =  ( SetCat ` 
ran  ( Hom f  `  C ) )
15 fvex 5812 . . . . . . 7  |-  ( Hom f  `  C )  e.  _V
1615rnex 6625 . . . . . 6  |-  ran  ( Hom f  `  C )  e.  _V
1716a1i 11 . . . . 5  |-  ( ph  ->  ran  ( Hom f  `  C )  e.  _V )
18 ssid 3486 . . . . . 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 15193 . . . 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 15166 . . 3  |-  ( ph  ->  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) )  =  (
<. D ,  (oppCat `  D ) >. curryF  (HomF
`  (oppCat `  C )
) ) )
223, 4, 9, 12hofpropd 15200 . . . 4  |-  ( ph  ->  (HomF
`  (oppCat `  C )
)  =  (HomF `  (oppCat `  D ) ) )
2322oveq2d 6219 . . 3  |-  ( ph  ->  ( <. D ,  (oppCat `  D ) >. curryF  (HomF
`  (oppCat `  C )
) )  =  (
<. D ,  (oppCat `  D ) >. curryF  (HomF
`  (oppCat `  D )
) ) )
2421, 23eqtrd 2495 . 2  |-  ( ph  ->  ( <. C ,  (oppCat `  C ) >. curryF  (HomF
`  (oppCat `  C )
) )  =  (
<. D ,  (oppCat `  D ) >. curryF  (HomF
`  (oppCat `  D )
) ) )
25 eqid 2454 . . 3  |-  (Yon `  C )  =  (Yon
`  C )
2625, 5, 7, 13yonval 15194 . 2  |-  ( ph  ->  (Yon `  C )  =  ( <. C , 
(oppCat `  C ) >. curryF  (HomF `  (oppCat `  C ) ) ) )
27 eqid 2454 . . 3  |-  (Yon `  D )  =  (Yon
`  D )
28 eqid 2454 . . 3  |-  (HomF `  (oppCat `  D ) )  =  (HomF
`  (oppCat `  D )
)
2927, 6, 10, 28yonval 15194 . 2  |-  ( ph  ->  (Yon `  D )  =  ( <. D , 
(oppCat `  D ) >. curryF  (HomF `  (oppCat `  D ) ) ) )
3024, 26, 293eqtr4d 2505 1  |-  ( ph  ->  (Yon `  C )  =  (Yon `  D )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3078    C_ wss 3439   <.cop 3994   ran crn 4952   ` cfv 5529  (class class class)co 6203   Catccat 14725   Hom f chomf 14727  compfccomf 14728  oppCatcoppc 14773   SetCatcsetc 15066   curryF ccurf 15143  HomFchof 15181  Yoncyon 15182
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-tpos 6858  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-10 10503  df-n0 10695  df-z 10762  df-dec 10871  df-uz 10977  df-fz 11559  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-hom 14385  df-cco 14386  df-cat 14729  df-cid 14730  df-homf 14731  df-comf 14732  df-oppc 14774  df-func 14891  df-setc 15067  df-xpc 15105  df-curf 15147  df-hof 15183  df-yon 15184
This theorem is referenced by: (None)
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