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Theorem oyoncl 15076
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 14657 . . . 4  |-  ( C  e.  Cat  ->  O  e.  Cat )
52, 4syl 16 . . 3  |-  ( ph  ->  O  e.  Cat )
6 eqid 2441 . . 3  |-  (oppCat `  O )  =  (oppCat `  O )
7 oyoncl.s . . 3  |-  S  =  ( SetCat `  U )
8 eqid 2441 . . 3  |-  ( (oppCat `  O ) FuncCat  S )  =  ( (oppCat `  O ) FuncCat  S )
9 oyoncl.u . . 3  |-  ( ph  ->  U  e.  V )
10 eqid 2441 . . . . . . 7  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
113, 10oppchomf 14655 . . . . . 6  |- tpos  ( Hom f  `  C )  =  ( Hom f  `  O )
1211rneqi 5062 . . . . 5  |-  ran tpos  ( Hom f  `  C )  =  ran  ( Hom f  `  O )
13 relxp 4943 . . . . . . 7  |-  Rel  (
( Base `  C )  X.  ( Base `  C
) )
14 eqid 2441 . . . . . . . . . 10  |-  ( Base `  C )  =  (
Base `  C )
1510, 14homffn 14628 . . . . . . . . 9  |-  ( Hom f  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
16 fndm 5507 . . . . . . . . 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 4919 . . . . . . 7  |-  ( Rel 
dom  ( Hom f  `  C )  <->  Rel  ( ( Base `  C
)  X.  ( Base `  C ) ) )
1913, 18mpbir 209 . . . . . 6  |-  Rel  dom  ( Hom f  `  C )
20 rntpos 6757 . . . . . 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 2463 . . . 4  |-  ran  ( Hom f  `  O )  =  ran  ( Hom f  `  C )
23 oyoncl.h . . . 4  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
2422, 23syl5eqss 3397 . . 3  |-  ( ph  ->  ran  ( Hom f  `  O ) 
C_  U )
251, 5, 6, 7, 8, 9, 24yoncl 15068 . 2  |-  ( ph  ->  Y  e.  ( O 
Func  ( (oppCat `  O ) FuncCat  S ) ) )
26 oyoncl.q . . . 4  |-  Q  =  ( C FuncCat  S )
2732oppchomf 14659 . . . . . 6  |-  ( Hom f  `  C )  =  ( Hom f  `  (oppCat `  O )
)
2827a1i 11 . . . . 5  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  (oppCat `  O
) ) )
2932oppccomf 14660 . . . . . 6  |-  (compf `  C
)  =  (compf `  (oppCat `  O ) )
3029a1i 11 . . . . 5  |-  ( ph  ->  (compf `  C )  =  (compf `  (oppCat `  O ) ) )
31 eqidd 2442 . . . . 5  |-  ( ph  ->  ( Hom f  `  S )  =  ( Hom f  `  S ) )
32 eqidd 2442 . . . . 5  |-  ( ph  ->  (compf `  S )  =  (compf `  S ) )
336oppccat 14657 . . . . . 6  |-  ( O  e.  Cat  ->  (oppCat `  O )  e.  Cat )
345, 33syl 16 . . . . 5  |-  ( ph  ->  (oppCat `  O )  e.  Cat )
357setccat 14949 . . . . . 6  |-  ( U  e.  V  ->  S  e.  Cat )
369, 35syl 16 . . . . 5  |-  ( ph  ->  S  e.  Cat )
3728, 30, 31, 32, 2, 34, 36, 36fucpropd 14883 . . . 4  |-  ( ph  ->  ( C FuncCat  S )  =  ( (oppCat `  O ) FuncCat  S ) )
3826, 37syl5eq 2485 . . 3  |-  ( ph  ->  Q  =  ( (oppCat `  O ) FuncCat  S ) )
3938oveq2d 6106 . 2  |-  ( ph  ->  ( O  Func  Q
)  =  ( O 
Func  ( (oppCat `  O ) FuncCat  S ) ) )
4025, 39eleqtrrd 2518 1  |-  ( ph  ->  Y  e.  ( O 
Func  Q ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1761    C_ wss 3325    X. cxp 4834   dom cdm 4836   ran crn 4837   Rel wrel 4841    Fn wfn 5410   ` cfv 5415  (class class class)co 6090  tpos ctpos 6743   Basecbs 14170   Catccat 14598   Hom f chomf 14600  compfccomf 14601  oppCatcoppc 14646    Func cfunc 14760   FuncCat cfuc 14848   SetCatcsetc 14939  Yoncyon 15055
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-hom 14258  df-cco 14259  df-cat 14602  df-cid 14603  df-homf 14604  df-comf 14605  df-oppc 14647  df-func 14764  df-nat 14849  df-fuc 14850  df-setc 14940  df-xpc 14978  df-curf 15020  df-hof 15056  df-yon 15057
This theorem is referenced by:  oyon1cl  15077
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