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Theorem oppchofcl 15184
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 2452 . . 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 14775 . . . 4  |-  ( C  e.  Cat  ->  O  e.  Cat )
74, 6syl 16 . . 3  |-  ( ph  ->  O  e.  Cat )
8 oppchofcl.u . . 3  |-  ( ph  ->  U  e.  V )
9 eqid 2452 . . . . . . 7  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
105, 9oppchomf 14773 . . . . . 6  |- tpos  ( Hom f  `  C )  =  ( Hom f  `  O )
1110rneqi 5169 . . . . 5  |-  ran tpos  ( Hom f  `  C )  =  ran  ( Hom f  `  O )
12 relxp 5050 . . . . . . 7  |-  Rel  (
( Base `  C )  X.  ( Base `  C
) )
13 eqid 2452 . . . . . . . . . 10  |-  ( Base `  C )  =  (
Base `  C )
149, 13homffn 14746 . . . . . . . . 9  |-  ( Hom f  `  C )  Fn  (
( Base `  C )  X.  ( Base `  C
) )
15 fndm 5613 . . . . . . . . 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 5026 . . . . . . 7  |-  ( Rel 
dom  ( Hom f  `  C )  <->  Rel  ( ( Base `  C
)  X.  ( Base `  C ) ) )
1812, 17mpbir 209 . . . . . 6  |-  Rel  dom  ( Hom f  `  C )
19 rntpos 6863 . . . . . 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 2483 . . . 4  |-  ran  ( Hom f  `  O )  =  ran  ( Hom f  `  C )
22 oppchofcl.h . . . 4  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
2321, 22syl5eqss 3503 . . 3  |-  ( ph  ->  ran  ( Hom f  `  O ) 
C_  U )
241, 2, 3, 7, 8, 23hofcl 15183 . 2  |-  ( ph  ->  M  e.  ( ( (oppCat `  O )  X.c  O )  Func  D
) )
2552oppchomf 14777 . . . . 5  |-  ( Hom f  `  C )  =  ( Hom f  `  (oppCat `  O )
)
2625a1i 11 . . . 4  |-  ( ph  ->  ( Hom f  `  C )  =  ( Hom f  `  (oppCat `  O
) ) )
2752oppccomf 14778 . . . . 5  |-  (compf `  C
)  =  (compf `  (oppCat `  O ) )
2827a1i 11 . . . 4  |-  ( ph  ->  (compf `  C )  =  (compf `  (oppCat `  O ) ) )
29 eqidd 2453 . . . 4  |-  ( ph  ->  ( Hom f  `  O )  =  ( Hom f  `  O ) )
30 eqidd 2453 . . . 4  |-  ( ph  ->  (compf `  O )  =  (compf `  O ) )
312oppccat 14775 . . . . 5  |-  ( O  e.  Cat  ->  (oppCat `  O )  e.  Cat )
327, 31syl 16 . . . 4  |-  ( ph  ->  (oppCat `  O )  e.  Cat )
3326, 28, 29, 30, 4, 32, 7, 7xpcpropd 15132 . . 3  |-  ( ph  ->  ( C  X.c  O )  =  ( (oppCat `  O )  X.c  O ) )
3433oveq1d 6210 . 2  |-  ( ph  ->  ( ( C  X.c  O
)  Func  D )  =  ( ( (oppCat `  O )  X.c  O ) 
Func  D ) )
3524, 34eleqtrrd 2543 1  |-  ( ph  ->  M  e.  ( ( C  X.c  O )  Func  D
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    C_ wss 3431    X. cxp 4941   dom cdm 4943   ran crn 4944   Rel wrel 4948    Fn wfn 5516   ` cfv 5521  (class class class)co 6195  tpos ctpos 6849   Basecbs 14287   Catccat 14716   Hom f chomf 14718  compfccomf 14719  oppCatcoppc 14764    Func cfunc 14878   SetCatcsetc 15057    X.c cxpc 15092  HomFchof 15172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-tpos 6850  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-map 7321  df-ixp 7369  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-n0 10686  df-z 10753  df-dec 10862  df-uz 10968  df-fz 11550  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-hom 14376  df-cco 14377  df-cat 14720  df-cid 14721  df-homf 14722  df-comf 14723  df-oppc 14765  df-func 14882  df-setc 15058  df-xpc 15096  df-hof 15174
This theorem is referenced by:  yoncl  15186  yon11  15188  yon12  15189  yon2  15190  yonpropd  15192
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