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Theorem oyoncl 16155
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 15627 . . . 4  |-  ( C  e.  Cat  ->  O  e.  Cat )
52, 4syl 17 . . 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 15625 . . . . . 6  |- tpos  ( Hom f  `  C )  =  ( Hom f  `  O )
1211rneqi 5061 . . . . 5  |-  ran tpos  ( Hom f  `  C )  =  ran  ( Hom f  `  O )
13 relxp 4942 . . . . . . 7  |-  Rel  (
( Base `  C )  X.  ( Base `  C
) )
14 eqid 2451 . . . . . . . . . 10  |-  ( Base `  C )  =  (
Base `  C )
1510, 14homffn 15598 . . . . . . . . 9  |-  ( Hom f  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
16 fndm 5675 . . . . . . . . 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 4918 . . . . . . 7  |-  ( Rel 
dom  ( Hom f  `  C )  <->  Rel  ( ( Base `  C
)  X.  ( Base `  C ) ) )
1913, 18mpbir 213 . . . . . 6  |-  Rel  dom  ( Hom f  `  C )
20 rntpos 6986 . . . . . 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 2475 . . . 4  |-  ran  ( Hom f  `  O )  =  ran  ( Hom f  `  C )
23 oyoncl.h . . . 4  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
2422, 23syl5eqss 3476 . . 3  |-  ( ph  ->  ran  ( Hom f  `  O ) 
C_  U )
251, 5, 6, 7, 8, 9, 24yoncl 16147 . 2  |-  ( ph  ->  Y  e.  ( O 
Func  ( (oppCat `  O ) FuncCat  S ) ) )
26 oyoncl.q . . . 4  |-  Q  =  ( C FuncCat  S )
2732oppchomf 15629 . . . . . 6  |-  ( Hom f  `  C )  =  ( Hom f  `  (oppCat `  O )
)
2827a1i 11 . . . . 5  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  (oppCat `  O
) ) )
2932oppccomf 15630 . . . . . 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 15627 . . . . . 6  |-  ( O  e.  Cat  ->  (oppCat `  O )  e.  Cat )
345, 33syl 17 . . . . 5  |-  ( ph  ->  (oppCat `  O )  e.  Cat )
357setccat 15980 . . . . . 6  |-  ( U  e.  V  ->  S  e.  Cat )
369, 35syl 17 . . . . 5  |-  ( ph  ->  S  e.  Cat )
3728, 30, 31, 32, 2, 34, 36, 36fucpropd 15882 . . . 4  |-  ( ph  ->  ( C FuncCat  S )  =  ( (oppCat `  O ) FuncCat  S ) )
3826, 37syl5eq 2497 . . 3  |-  ( ph  ->  Q  =  ( (oppCat `  O ) FuncCat  S ) )
3938oveq2d 6306 . 2  |-  ( ph  ->  ( O  Func  Q
)  =  ( O 
Func  ( (oppCat `  O ) FuncCat  S ) ) )
4025, 39eleqtrrd 2532 1  |-  ( ph  ->  Y  e.  ( O 
Func  Q ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1444    e. wcel 1887    C_ wss 3404    X. cxp 4832   dom cdm 4834   ran crn 4835   Rel wrel 4839    Fn wfn 5577   ` cfv 5582  (class class class)co 6290  tpos ctpos 6972   Basecbs 15121   Catccat 15570   Hom f chomf 15572  compfccomf 15573  oppCatcoppc 15616    Func cfunc 15759   FuncCat cfuc 15847   SetCatcsetc 15970  Yoncyon 16134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-tpos 6973  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11785  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-hom 15214  df-cco 15215  df-cat 15574  df-cid 15575  df-homf 15576  df-comf 15577  df-oppc 15617  df-func 15763  df-nat 15848  df-fuc 15849  df-setc 15971  df-xpc 16057  df-curf 16099  df-hof 16135  df-yon 16136
This theorem is referenced by:  oyon1cl  16156
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