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Theorem oyoncl 15182
Description: The opposite Yoneda embedding is a functor from oppCat `  C to the functor category  C  ->  SetCat. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
oyoncl.o  |-  O  =  (oppCat `  C )
oyoncl.y  |-  Y  =  (Yon `  O )
oyoncl.c  |-  ( ph  ->  C  e.  Cat )
oyoncl.s  |-  S  =  ( SetCat `  U )
oyoncl.u  |-  ( ph  ->  U  e.  V )
oyoncl.h  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
oyoncl.q  |-  Q  =  ( C FuncCat  S )
Assertion
Ref Expression
oyoncl  |-  ( ph  ->  Y  e.  ( O 
Func  Q ) )

Proof of Theorem oyoncl
StepHypRef Expression
1 oyoncl.y . . 3  |-  Y  =  (Yon `  O )
2 oyoncl.c . . . 4  |-  ( ph  ->  C  e.  Cat )
3 oyoncl.o . . . . 5  |-  O  =  (oppCat `  C )
43oppccat 14763 . . . 4  |-  ( C  e.  Cat  ->  O  e.  Cat )
52, 4syl 16 . . 3  |-  ( ph  ->  O  e.  Cat )
6 eqid 2451 . . 3  |-  (oppCat `  O )  =  (oppCat `  O )
7 oyoncl.s . . 3  |-  S  =  ( SetCat `  U )
8 eqid 2451 . . 3  |-  ( (oppCat `  O ) FuncCat  S )  =  ( (oppCat `  O ) FuncCat  S )
9 oyoncl.u . . 3  |-  ( ph  ->  U  e.  V )
10 eqid 2451 . . . . . . 7  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
113, 10oppchomf 14761 . . . . . 6  |- tpos  ( Hom f  `  C )  =  ( Hom f  `  O )
1211rneqi 5164 . . . . 5  |-  ran tpos  ( Hom f  `  C )  =  ran  ( Hom f  `  O )
13 relxp 5045 . . . . . . 7  |-  Rel  (
( Base `  C )  X.  ( Base `  C
) )
14 eqid 2451 . . . . . . . . . 10  |-  ( Base `  C )  =  (
Base `  C )
1510, 14homffn 14734 . . . . . . . . 9  |-  ( Hom f  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
16 fndm 5608 . . . . . . . . 9  |-  ( ( Hom f  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )  ->  dom  ( Hom f  `  C )  =  ( ( Base `  C
)  X.  ( Base `  C ) ) )
1715, 16ax-mp 5 . . . . . . . 8  |-  dom  ( Hom f  `  C )  =  ( ( Base `  C
)  X.  ( Base `  C ) )
1817releqi 5021 . . . . . . 7  |-  ( Rel 
dom  ( Hom f  `  C )  <->  Rel  ( ( Base `  C
)  X.  ( Base `  C ) ) )
1913, 18mpbir 209 . . . . . 6  |-  Rel  dom  ( Hom f  `  C )
20 rntpos 6858 . . . . . 6  |-  ( Rel 
dom  ( Hom f  `  C )  ->  ran tpos  ( Hom f  `  C
)  =  ran  ( Hom f  `  C ) )
2119, 20ax-mp 5 . . . . 5  |-  ran tpos  ( Hom f  `  C )  =  ran  ( Hom f  `  C )
2212, 21eqtr3i 2482 . . . 4  |-  ran  ( Hom f  `  O )  =  ran  ( Hom f  `  C )
23 oyoncl.h . . . 4  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
2422, 23syl5eqss 3498 . . 3  |-  ( ph  ->  ran  ( Hom f  `  O ) 
C_  U )
251, 5, 6, 7, 8, 9, 24yoncl 15174 . 2  |-  ( ph  ->  Y  e.  ( O 
Func  ( (oppCat `  O ) FuncCat  S ) ) )
26 oyoncl.q . . . 4  |-  Q  =  ( C FuncCat  S )
2732oppchomf 14765 . . . . . 6  |-  ( Hom f  `  C )  =  ( Hom f  `  (oppCat `  O )
)
2827a1i 11 . . . . 5  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  (oppCat `  O
) ) )
2932oppccomf 14766 . . . . . 6  |-  (compf `  C
)  =  (compf `  (oppCat `  O ) )
3029a1i 11 . . . . 5  |-  ( ph  ->  (compf `  C )  =  (compf `  (oppCat `  O ) ) )
31 eqidd 2452 . . . . 5  |-  ( ph  ->  ( Hom f  `  S )  =  ( Hom f  `  S ) )
32 eqidd 2452 . . . . 5  |-  ( ph  ->  (compf `  S )  =  (compf `  S ) )
336oppccat 14763 . . . . . 6  |-  ( O  e.  Cat  ->  (oppCat `  O )  e.  Cat )
345, 33syl 16 . . . . 5  |-  ( ph  ->  (oppCat `  O )  e.  Cat )
357setccat 15055 . . . . . 6  |-  ( U  e.  V  ->  S  e.  Cat )
369, 35syl 16 . . . . 5  |-  ( ph  ->  S  e.  Cat )
3728, 30, 31, 32, 2, 34, 36, 36fucpropd 14989 . . . 4  |-  ( ph  ->  ( C FuncCat  S )  =  ( (oppCat `  O ) FuncCat  S ) )
3826, 37syl5eq 2504 . . 3  |-  ( ph  ->  Q  =  ( (oppCat `  O ) FuncCat  S ) )
3938oveq2d 6206 . 2  |-  ( ph  ->  ( O  Func  Q
)  =  ( O 
Func  ( (oppCat `  O ) FuncCat  S ) ) )
4025, 39eleqtrrd 2542 1  |-  ( ph  ->  Y  e.  ( O 
Func  Q ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    C_ wss 3426    X. cxp 4936   dom cdm 4938   ran crn 4939   Rel wrel 4943    Fn wfn 5511   ` cfv 5516  (class class class)co 6190  tpos ctpos 6844   Basecbs 14276   Catccat 14704   Hom f chomf 14706  compfccomf 14707  oppCatcoppc 14752    Func cfunc 14866   FuncCat cfuc 14954   SetCatcsetc 15045  Yoncyon 15161
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 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
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 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-tpos 6845  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-fz 11539  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-hom 14364  df-cco 14365  df-cat 14708  df-cid 14709  df-homf 14710  df-comf 14711  df-oppc 14753  df-func 14870  df-nat 14955  df-fuc 14956  df-setc 15046  df-xpc 15084  df-curf 15126  df-hof 15162  df-yon 15163
This theorem is referenced by:  oyon1cl  15183
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