MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oppchofcl Unicode version

Theorem oppchofcl 14050
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 2296 . . 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 13641 . . . 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 2296 . . . . . . 7  |-  (  Homf  `  C )  =  (  Homf 
`  C )
105, 9oppchomf 13639 . . . . . 6  |- tpos  (  Homf  `  C )  =  (  Homf 
`  O )
1110rneqi 4921 . . . . 5  |-  ran tpos  (  Homf  `  C )  =  ran  (  Homf 
`  O )
12 relxp 4810 . . . . . . 7  |-  Rel  (
( Base `  C )  X.  ( Base `  C
) )
13 eqid 2296 . . . . . . . . . 10  |-  ( Base `  C )  =  (
Base `  C )
149, 13homffn 13612 . . . . . . . . 9  |-  (  Homf  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
15 fndm 5359 . . . . . . . . 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 4788 . . . . . . 7  |-  ( Rel 
dom  (  Homf  `  C )  <->  Rel  ( ( Base `  C
)  X.  ( Base `  C ) ) )
1812, 17mpbir 200 . . . . . 6  |-  Rel  dom  (  Homf 
`  C )
19 rntpos 6263 . . . . . 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 2318 . . . 4  |-  ran  (  Homf  `  O )  =  ran  (  Homf 
`  C )
22 oppchofcl.h . . . 4  |-  ( ph  ->  ran  (  Homf  `  C ) 
C_  U )
2321, 22syl5eqss 3235 . . 3  |-  ( ph  ->  ran  (  Homf  `  O ) 
C_  U )
241, 2, 3, 7, 8, 23hofcl 14049 . 2  |-  ( ph  ->  M  e.  ( ( (oppCat `  O )  X.c  O )  Func  D
) )
2552oppchomf 13643 . . . . 5  |-  (  Homf  `  C )  =  (  Homf 
`  (oppCat `  O )
)
2625a1i 10 . . . 4  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  (oppCat `  O )
) )
2752oppccomf 13644 . . . . 5  |-  (compf `  C
)  =  (compf `  (oppCat `  O ) )
2827a1i 10 . . . 4  |-  ( ph  ->  (compf `  C )  =  (compf `  (oppCat `  O ) ) )
29 eqidd 2297 . . . 4  |-  ( ph  ->  (  Homf 
`  O )  =  (  Homf 
`  O ) )
30 eqidd 2297 . . . 4  |-  ( ph  ->  (compf `  O )  =  (compf `  O ) )
312oppccat 13641 . . . . 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 13998 . . 3  |-  ( ph  ->  ( C  X.c  O )  =  ( (oppCat `  O )  X.c  O ) )
3433oveq1d 5889 . 2  |-  ( ph  ->  ( ( C  X.c  O
)  Func  D )  =  ( ( (oppCat `  O )  X.c  O ) 
Func  D ) )
3524, 34eleqtrrd 2373 1  |-  ( ph  ->  M  e.  ( ( C  X.c  O )  Func  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    C_ wss 3165    X. cxp 4703   dom cdm 4705   ran crn 4706   Rel wrel 4710    Fn wfn 5266   ` cfv 5271  (class class class)co 5874  tpos ctpos 6249   Basecbs 13164   Catccat 13582    Homf chomf 13584  compfccomf 13585  oppCatcoppc 13630    Func cfunc 13744   SetCatcsetc 13923    X.c cxpc 13958  HomFchof 14038
This theorem is referenced by:  yoncl  14052  yon11  14054  yon12  14055  yon2  14056  yonpropd  14058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-hom 13248  df-cco 13249  df-cat 13586  df-cid 13587  df-homf 13588  df-comf 13589  df-oppc 13631  df-func 13748  df-setc 13924  df-xpc 13962  df-hof 14040
  Copyright terms: Public domain W3C validator