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Theorem oppchofcl 16138
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 2450 . . 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 15620 . . . 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 2450 . . . . . . 7  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
105, 9oppchomf 15618 . . . . . 6  |- tpos  ( Hom f  `  C )  =  ( Hom f  `  O )
1110rneqi 5060 . . . . 5  |-  ran tpos  ( Hom f  `  C )  =  ran  ( Hom f  `  O )
12 relxp 4941 . . . . . . 7  |-  Rel  (
( Base `  C )  X.  ( Base `  C
) )
13 eqid 2450 . . . . . . . . . 10  |-  ( Base `  C )  =  (
Base `  C )
149, 13homffn 15591 . . . . . . . . 9  |-  ( Hom f  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
15 fndm 5673 . . . . . . . . 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 4917 . . . . . . 7  |-  ( Rel 
dom  ( Hom f  `  C )  <->  Rel  ( ( Base `  C
)  X.  ( Base `  C ) ) )
1812, 17mpbir 213 . . . . . 6  |-  Rel  dom  ( Hom f  `  C )
19 rntpos 6983 . . . . . 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 2474 . . . 4  |-  ran  ( Hom f  `  O )  =  ran  ( Hom f  `  C )
22 oppchofcl.h . . . 4  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
2321, 22syl5eqss 3475 . . 3  |-  ( ph  ->  ran  ( Hom f  `  O ) 
C_  U )
241, 2, 3, 7, 8, 23hofcl 16137 . 2  |-  ( ph  ->  M  e.  ( ( (oppCat `  O )  X.c  O )  Func  D
) )
2552oppchomf 15622 . . . . 5  |-  ( Hom f  `  C )  =  ( Hom f  `  (oppCat `  O )
)
2625a1i 11 . . . 4  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  (oppCat `  O
) ) )
2752oppccomf 15623 . . . . 5  |-  (compf `  C
)  =  (compf `  (oppCat `  O ) )
2827a1i 11 . . . 4  |-  ( ph  ->  (compf `  C )  =  (compf `  (oppCat `  O ) ) )
29 eqidd 2451 . . . 4  |-  ( ph  ->  ( Hom f  `  O )  =  ( Hom f  `  O ) )
30 eqidd 2451 . . . 4  |-  ( ph  ->  (compf `  O )  =  (compf `  O ) )
312oppccat 15620 . . . . 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 16086 . . 3  |-  ( ph  ->  ( C  X.c  O )  =  ( (oppCat `  O )  X.c  O ) )
3433oveq1d 6303 . 2  |-  ( ph  ->  ( ( C  X.c  O
)  Func  D )  =  ( ( (oppCat `  O )  X.c  O ) 
Func  D ) )
3524, 34eleqtrrd 2531 1  |-  ( ph  ->  M  e.  ( ( C  X.c  O )  Func  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1443    e. wcel 1886    C_ wss 3403    X. cxp 4831   dom cdm 4833   ran crn 4834   Rel wrel 4838    Fn wfn 5576   ` cfv 5581  (class class class)co 6288  tpos ctpos 6969   Basecbs 15114   Catccat 15563   Hom f chomf 15565  compfccomf 15566  oppCatcoppc 15609    Func cfunc 15752   SetCatcsetc 15963    X.c cxpc 16046  HomFchof 16126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-tpos 6970  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-fz 11782  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-hom 15207  df-cco 15208  df-cat 15567  df-cid 15568  df-homf 15569  df-comf 15570  df-oppc 15610  df-func 15756  df-setc 15964  df-xpc 16050  df-hof 16128
This theorem is referenced by:  yoncl  16140  yon11  16142  yon12  16143  yon2  16144  yonpropd  16146
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