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Theorem oppchofcl 16096
Description: Closure of the opposite Hom functor. (Contributed by Mario Carneiro, 17-Jan-2017.)
Hypotheses
Ref Expression
oppchofcl.o  |-  O  =  (oppCat `  C )
oppchofcl.m  |-  M  =  (HomF
`  O )
oppchofcl.d  |-  D  =  ( SetCat `  U )
oppchofcl.c  |-  ( ph  ->  C  e.  Cat )
oppchofcl.u  |-  ( ph  ->  U  e.  V )
oppchofcl.h  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
Assertion
Ref Expression
oppchofcl  |-  ( ph  ->  M  e.  ( ( C  X.c  O )  Func  D
) )

Proof of Theorem oppchofcl
StepHypRef Expression
1 oppchofcl.m . . 3  |-  M  =  (HomF
`  O )
2 eqid 2429 . . 3  |-  (oppCat `  O )  =  (oppCat `  O )
3 oppchofcl.d . . 3  |-  D  =  ( SetCat `  U )
4 oppchofcl.c . . . 4  |-  ( ph  ->  C  e.  Cat )
5 oppchofcl.o . . . . 5  |-  O  =  (oppCat `  C )
65oppccat 15578 . . . 4  |-  ( C  e.  Cat  ->  O  e.  Cat )
74, 6syl 17 . . 3  |-  ( ph  ->  O  e.  Cat )
8 oppchofcl.u . . 3  |-  ( ph  ->  U  e.  V )
9 eqid 2429 . . . . . . 7  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
105, 9oppchomf 15576 . . . . . 6  |- tpos  ( Hom f  `  C )  =  ( Hom f  `  O )
1110rneqi 5081 . . . . 5  |-  ran tpos  ( Hom f  `  C )  =  ran  ( Hom f  `  O )
12 relxp 4962 . . . . . . 7  |-  Rel  (
( Base `  C )  X.  ( Base `  C
) )
13 eqid 2429 . . . . . . . . . 10  |-  ( Base `  C )  =  (
Base `  C )
149, 13homffn 15549 . . . . . . . . 9  |-  ( Hom f  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
15 fndm 5693 . . . . . . . . 9  |-  ( ( Hom f  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )  ->  dom  ( Hom f  `  C )  =  ( ( Base `  C
)  X.  ( Base `  C ) ) )
1614, 15ax-mp 5 . . . . . . . 8  |-  dom  ( Hom f  `  C )  =  ( ( Base `  C
)  X.  ( Base `  C ) )
1716releqi 4938 . . . . . . 7  |-  ( Rel 
dom  ( Hom f  `  C )  <->  Rel  ( ( Base `  C
)  X.  ( Base `  C ) ) )
1812, 17mpbir 212 . . . . . 6  |-  Rel  dom  ( Hom f  `  C )
19 rntpos 6994 . . . . . 6  |-  ( Rel 
dom  ( Hom f  `  C )  ->  ran tpos  ( Hom f  `  C
)  =  ran  ( Hom f  `  C ) )
2018, 19ax-mp 5 . . . . 5  |-  ran tpos  ( Hom f  `  C )  =  ran  ( Hom f  `  C )
2111, 20eqtr3i 2460 . . . 4  |-  ran  ( Hom f  `  O )  =  ran  ( Hom f  `  C )
22 oppchofcl.h . . . 4  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
2321, 22syl5eqss 3514 . . 3  |-  ( ph  ->  ran  ( Hom f  `  O ) 
C_  U )
241, 2, 3, 7, 8, 23hofcl 16095 . 2  |-  ( ph  ->  M  e.  ( ( (oppCat `  O )  X.c  O )  Func  D
) )
2552oppchomf 15580 . . . . 5  |-  ( Hom f  `  C )  =  ( Hom f  `  (oppCat `  O )
)
2625a1i 11 . . . 4  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  (oppCat `  O
) ) )
2752oppccomf 15581 . . . . 5  |-  (compf `  C
)  =  (compf `  (oppCat `  O ) )
2827a1i 11 . . . 4  |-  ( ph  ->  (compf `  C )  =  (compf `  (oppCat `  O ) ) )
29 eqidd 2430 . . . 4  |-  ( ph  ->  ( Hom f  `  O )  =  ( Hom f  `  O ) )
30 eqidd 2430 . . . 4  |-  ( ph  ->  (compf `  O )  =  (compf `  O ) )
312oppccat 15578 . . . . 5  |-  ( O  e.  Cat  ->  (oppCat `  O )  e.  Cat )
327, 31syl 17 . . . 4  |-  ( ph  ->  (oppCat `  O )  e.  Cat )
3326, 28, 29, 30, 4, 32, 7, 7xpcpropd 16044 . . 3  |-  ( ph  ->  ( C  X.c  O )  =  ( (oppCat `  O )  X.c  O ) )
3433oveq1d 6320 . 2  |-  ( ph  ->  ( ( C  X.c  O
)  Func  D )  =  ( ( (oppCat `  O )  X.c  O ) 
Func  D ) )
3524, 34eleqtrrd 2520 1  |-  ( ph  ->  M  e.  ( ( C  X.c  O )  Func  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870    C_ wss 3442    X. cxp 4852   dom cdm 4854   ran crn 4855   Rel wrel 4859    Fn wfn 5596   ` cfv 5601  (class class class)co 6305  tpos ctpos 6980   Basecbs 15084   Catccat 15521   Hom f chomf 15523  compfccomf 15524  oppCatcoppc 15567    Func cfunc 15710   SetCatcsetc 15921    X.c cxpc 16004  HomFchof 16084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  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 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-hom 15176  df-cco 15177  df-cat 15525  df-cid 15526  df-homf 15527  df-comf 15528  df-oppc 15568  df-func 15714  df-setc 15922  df-xpc 16008  df-hof 16086
This theorem is referenced by:  yoncl  16098  yon11  16100  yon12  16101  yon2  16102  yonpropd  16104
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