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Theorem oyoncl 14367
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 13948 . . . 4  |-  ( C  e.  Cat  ->  O  e.  Cat )
52, 4syl 16 . . 3  |-  ( ph  ->  O  e.  Cat )
6 eqid 2436 . . 3  |-  (oppCat `  O )  =  (oppCat `  O )
7 oyoncl.s . . 3  |-  S  =  ( SetCat `  U )
8 eqid 2436 . . 3  |-  ( (oppCat `  O ) FuncCat  S )  =  ( (oppCat `  O ) FuncCat  S )
9 oyoncl.u . . 3  |-  ( ph  ->  U  e.  V )
10 eqid 2436 . . . . . . 7  |-  (  Homf  `  C )  =  (  Homf 
`  C )
113, 10oppchomf 13946 . . . . . 6  |- tpos  (  Homf  `  C )  =  (  Homf 
`  O )
1211rneqi 5096 . . . . 5  |-  ran tpos  (  Homf  `  C )  =  ran  (  Homf 
`  O )
13 relxp 4983 . . . . . . 7  |-  Rel  (
( Base `  C )  X.  ( Base `  C
) )
14 eqid 2436 . . . . . . . . . 10  |-  ( Base `  C )  =  (
Base `  C )
1510, 14homffn 13919 . . . . . . . . 9  |-  (  Homf  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
16 fndm 5544 . . . . . . . . 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 4960 . . . . . . 7  |-  ( Rel 
dom  (  Homf  `  C )  <->  Rel  ( ( Base `  C
)  X.  ( Base `  C ) ) )
1913, 18mpbir 201 . . . . . 6  |-  Rel  dom  (  Homf 
`  C )
20 rntpos 6492 . . . . . 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 2458 . . . 4  |-  ran  (  Homf  `  O )  =  ran  (  Homf 
`  C )
23 oyoncl.h . . . 4  |-  ( ph  ->  ran  (  Homf  `  C ) 
C_  U )
2422, 23syl5eqss 3392 . . 3  |-  ( ph  ->  ran  (  Homf  `  O ) 
C_  U )
251, 5, 6, 7, 8, 9, 24yoncl 14359 . 2  |-  ( ph  ->  Y  e.  ( O 
Func  ( (oppCat `  O ) FuncCat  S ) ) )
26 oyoncl.q . . . 4  |-  Q  =  ( C FuncCat  S )
2732oppchomf 13950 . . . . . 6  |-  (  Homf  `  C )  =  (  Homf 
`  (oppCat `  O )
)
2827a1i 11 . . . . 5  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  (oppCat `  O )
) )
2932oppccomf 13951 . . . . . 6  |-  (compf `  C
)  =  (compf `  (oppCat `  O ) )
3029a1i 11 . . . . 5  |-  ( ph  ->  (compf `  C )  =  (compf `  (oppCat `  O ) ) )
31 eqidd 2437 . . . . 5  |-  ( ph  ->  (  Homf 
`  S )  =  (  Homf 
`  S ) )
32 eqidd 2437 . . . . 5  |-  ( ph  ->  (compf `  S )  =  (compf `  S ) )
336oppccat 13948 . . . . . 6  |-  ( O  e.  Cat  ->  (oppCat `  O )  e.  Cat )
345, 33syl 16 . . . . 5  |-  ( ph  ->  (oppCat `  O )  e.  Cat )
357setccat 14240 . . . . . 6  |-  ( U  e.  V  ->  S  e.  Cat )
369, 35syl 16 . . . . 5  |-  ( ph  ->  S  e.  Cat )
3728, 30, 31, 32, 2, 34, 36, 36fucpropd 14174 . . . 4  |-  ( ph  ->  ( C FuncCat  S )  =  ( (oppCat `  O ) FuncCat  S ) )
3826, 37syl5eq 2480 . . 3  |-  ( ph  ->  Q  =  ( (oppCat `  O ) FuncCat  S ) )
3938oveq2d 6097 . 2  |-  ( ph  ->  ( O  Func  Q
)  =  ( O 
Func  ( (oppCat `  O ) FuncCat  S ) ) )
4025, 39eleqtrrd 2513 1  |-  ( ph  ->  Y  e.  ( O 
Func  Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    C_ wss 3320    X. cxp 4876   dom cdm 4878   ran crn 4879   Rel wrel 4883    Fn wfn 5449   ` cfv 5454  (class class class)co 6081  tpos ctpos 6478   Basecbs 13469   Catccat 13889    Homf chomf 13891  compfccomf 13892  oppCatcoppc 13937    Func cfunc 14051   FuncCat cfuc 14139   SetCatcsetc 14230  Yoncyon 14346
This theorem is referenced by:  oyon1cl  14368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-hom 13553  df-cco 13554  df-cat 13893  df-cid 13894  df-homf 13895  df-comf 13896  df-oppc 13938  df-func 14055  df-nat 14140  df-fuc 14141  df-setc 14231  df-xpc 14269  df-curf 14311  df-hof 14347  df-yon 14348
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