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Theorem oppchofcl 14034
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  (  Homf  `  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 2283 . . 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 13625 . . . 4  |-  ( C  e.  Cat  ->  O  e.  Cat )
74, 6syl 15 . . 3  |-  ( ph  ->  O  e.  Cat )
8 oppchofcl.u . . 3  |-  ( ph  ->  U  e.  V )
9 eqid 2283 . . . . . . 7  |-  (  Homf  `  C )  =  (  Homf 
`  C )
105, 9oppchomf 13623 . . . . . 6  |- tpos  (  Homf  `  C )  =  (  Homf 
`  O )
1110rneqi 4905 . . . . 5  |-  ran tpos  (  Homf  `  C )  =  ran  (  Homf 
`  O )
12 relxp 4794 . . . . . . 7  |-  Rel  (
( Base `  C )  X.  ( Base `  C
) )
13 eqid 2283 . . . . . . . . . 10  |-  ( Base `  C )  =  (
Base `  C )
149, 13homffn 13596 . . . . . . . . 9  |-  (  Homf  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
15 fndm 5343 . . . . . . . . 9  |-  ( (  Homf 
`  C )  Fn  ( ( Base `  C
)  X.  ( Base `  C ) )  ->  dom  (  Homf 
`  C )  =  ( ( Base `  C
)  X.  ( Base `  C ) ) )
1614, 15ax-mp 8 . . . . . . . 8  |-  dom  (  Homf  `  C )  =  ( ( Base `  C
)  X.  ( Base `  C ) )
1716releqi 4772 . . . . . . 7  |-  ( Rel 
dom  (  Homf  `  C )  <->  Rel  ( ( Base `  C
)  X.  ( Base `  C ) ) )
1812, 17mpbir 200 . . . . . 6  |-  Rel  dom  (  Homf 
`  C )
19 rntpos 6247 . . . . . 6  |-  ( Rel 
dom  (  Homf  `  C )  ->  ran tpos  (  Homf  `  C )  =  ran  (  Homf  `  C ) )
2018, 19ax-mp 8 . . . . 5  |-  ran tpos  (  Homf  `  C )  =  ran  (  Homf 
`  C )
2111, 20eqtr3i 2305 . . . 4  |-  ran  (  Homf  `  O )  =  ran  (  Homf 
`  C )
22 oppchofcl.h . . . 4  |-  ( ph  ->  ran  (  Homf  `  C ) 
C_  U )
2321, 22syl5eqss 3222 . . 3  |-  ( ph  ->  ran  (  Homf  `  O ) 
C_  U )
241, 2, 3, 7, 8, 23hofcl 14033 . 2  |-  ( ph  ->  M  e.  ( ( (oppCat `  O )  X.c  O )  Func  D
) )
2552oppchomf 13627 . . . . 5  |-  (  Homf  `  C )  =  (  Homf 
`  (oppCat `  O )
)
2625a1i 10 . . . 4  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  (oppCat `  O )
) )
2752oppccomf 13628 . . . . 5  |-  (compf `  C
)  =  (compf `  (oppCat `  O ) )
2827a1i 10 . . . 4  |-  ( ph  ->  (compf `  C )  =  (compf `  (oppCat `  O ) ) )
29 eqidd 2284 . . . 4  |-  ( ph  ->  (  Homf 
`  O )  =  (  Homf 
`  O ) )
30 eqidd 2284 . . . 4  |-  ( ph  ->  (compf `  O )  =  (compf `  O ) )
312oppccat 13625 . . . . 5  |-  ( O  e.  Cat  ->  (oppCat `  O )  e.  Cat )
327, 31syl 15 . . . 4  |-  ( ph  ->  (oppCat `  O )  e.  Cat )
3326, 28, 29, 30, 4, 32, 7, 7xpcpropd 13982 . . 3  |-  ( ph  ->  ( C  X.c  O )  =  ( (oppCat `  O )  X.c  O ) )
3433oveq1d 5873 . 2  |-  ( ph  ->  ( ( C  X.c  O
)  Func  D )  =  ( ( (oppCat `  O )  X.c  O ) 
Func  D ) )
3524, 34eleqtrrd 2360 1  |-  ( ph  ->  M  e.  ( ( C  X.c  O )  Func  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    C_ wss 3152    X. cxp 4687   dom cdm 4689   ran crn 4690   Rel wrel 4694    Fn wfn 5250   ` cfv 5255  (class class class)co 5858  tpos ctpos 6233   Basecbs 13148   Catccat 13566    Homf chomf 13568  compfccomf 13569  oppCatcoppc 13614    Func cfunc 13728   SetCatcsetc 13907    X.c cxpc 13942  HomFchof 14022
This theorem is referenced by:  yoncl  14036  yon11  14038  yon12  14039  yon2  14040  yonpropd  14042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-hom 13232  df-cco 13233  df-cat 13570  df-cid 13571  df-homf 13572  df-comf 13573  df-oppc 13615  df-func 13732  df-setc 13908  df-xpc 13946  df-hof 14024
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