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Theorem oyoncl 14093
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  (  Homf  `  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 13674 . . . 4  |-  ( C  e.  Cat  ->  O  e.  Cat )
52, 4syl 15 . . 3  |-  ( ph  ->  O  e.  Cat )
6 eqid 2316 . . 3  |-  (oppCat `  O )  =  (oppCat `  O )
7 oyoncl.s . . 3  |-  S  =  ( SetCat `  U )
8 eqid 2316 . . 3  |-  ( (oppCat `  O ) FuncCat  S )  =  ( (oppCat `  O ) FuncCat  S )
9 oyoncl.u . . 3  |-  ( ph  ->  U  e.  V )
10 eqid 2316 . . . . . . 7  |-  (  Homf  `  C )  =  (  Homf 
`  C )
113, 10oppchomf 13672 . . . . . 6  |- tpos  (  Homf  `  C )  =  (  Homf 
`  O )
1211rneqi 4942 . . . . 5  |-  ran tpos  (  Homf  `  C )  =  ran  (  Homf 
`  O )
13 relxp 4831 . . . . . . 7  |-  Rel  (
( Base `  C )  X.  ( Base `  C
) )
14 eqid 2316 . . . . . . . . . 10  |-  ( Base `  C )  =  (
Base `  C )
1510, 14homffn 13645 . . . . . . . . 9  |-  (  Homf  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
16 fndm 5380 . . . . . . . . 9  |-  ( (  Homf 
`  C )  Fn  ( ( Base `  C
)  X.  ( Base `  C ) )  ->  dom  (  Homf 
`  C )  =  ( ( Base `  C
)  X.  ( Base `  C ) ) )
1715, 16ax-mp 8 . . . . . . . 8  |-  dom  (  Homf  `  C )  =  ( ( Base `  C
)  X.  ( Base `  C ) )
1817releqi 4809 . . . . . . 7  |-  ( Rel 
dom  (  Homf  `  C )  <->  Rel  ( ( Base `  C
)  X.  ( Base `  C ) ) )
1913, 18mpbir 200 . . . . . 6  |-  Rel  dom  (  Homf 
`  C )
20 rntpos 6289 . . . . . 6  |-  ( Rel 
dom  (  Homf  `  C )  ->  ran tpos  (  Homf  `  C )  =  ran  (  Homf  `  C ) )
2119, 20ax-mp 8 . . . . 5  |-  ran tpos  (  Homf  `  C )  =  ran  (  Homf 
`  C )
2212, 21eqtr3i 2338 . . . 4  |-  ran  (  Homf  `  O )  =  ran  (  Homf 
`  C )
23 oyoncl.h . . . 4  |-  ( ph  ->  ran  (  Homf  `  C ) 
C_  U )
2422, 23syl5eqss 3256 . . 3  |-  ( ph  ->  ran  (  Homf  `  O ) 
C_  U )
251, 5, 6, 7, 8, 9, 24yoncl 14085 . 2  |-  ( ph  ->  Y  e.  ( O 
Func  ( (oppCat `  O ) FuncCat  S ) ) )
26 oyoncl.q . . . 4  |-  Q  =  ( C FuncCat  S )
2732oppchomf 13676 . . . . . 6  |-  (  Homf  `  C )  =  (  Homf 
`  (oppCat `  O )
)
2827a1i 10 . . . . 5  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  (oppCat `  O )
) )
2932oppccomf 13677 . . . . . 6  |-  (compf `  C
)  =  (compf `  (oppCat `  O ) )
3029a1i 10 . . . . 5  |-  ( ph  ->  (compf `  C )  =  (compf `  (oppCat `  O ) ) )
31 eqidd 2317 . . . . 5  |-  ( ph  ->  (  Homf 
`  S )  =  (  Homf 
`  S ) )
32 eqidd 2317 . . . . 5  |-  ( ph  ->  (compf `  S )  =  (compf `  S ) )
336oppccat 13674 . . . . . 6  |-  ( O  e.  Cat  ->  (oppCat `  O )  e.  Cat )
345, 33syl 15 . . . . 5  |-  ( ph  ->  (oppCat `  O )  e.  Cat )
357setccat 13966 . . . . . 6  |-  ( U  e.  V  ->  S  e.  Cat )
369, 35syl 15 . . . . 5  |-  ( ph  ->  S  e.  Cat )
3728, 30, 31, 32, 2, 34, 36, 36fucpropd 13900 . . . 4  |-  ( ph  ->  ( C FuncCat  S )  =  ( (oppCat `  O ) FuncCat  S ) )
3826, 37syl5eq 2360 . . 3  |-  ( ph  ->  Q  =  ( (oppCat `  O ) FuncCat  S ) )
3938oveq2d 5916 . 2  |-  ( ph  ->  ( O  Func  Q
)  =  ( O 
Func  ( (oppCat `  O ) FuncCat  S ) ) )
4025, 39eleqtrrd 2393 1  |-  ( ph  ->  Y  e.  ( O 
Func  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701    C_ wss 3186    X. cxp 4724   dom cdm 4726   ran crn 4727   Rel wrel 4731    Fn wfn 5287   ` cfv 5292  (class class class)co 5900  tpos ctpos 6275   Basecbs 13195   Catccat 13615    Homf chomf 13617  compfccomf 13618  oppCatcoppc 13663    Func cfunc 13777   FuncCat cfuc 13865   SetCatcsetc 13956  Yoncyon 14072
This theorem is referenced by:  oyon1cl  14094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-tpos 6276  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-ixp 6861  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-fz 10830  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-hom 13279  df-cco 13280  df-cat 13619  df-cid 13620  df-homf 13621  df-comf 13622  df-oppc 13664  df-func 13781  df-nat 13866  df-fuc 13867  df-setc 13957  df-xpc 13995  df-curf 14037  df-hof 14073  df-yon 14074
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