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Theorem fulloppc 14120
Description: The opposite functor of a full functor is also full. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
fulloppc.o  |-  O  =  (oppCat `  C )
fulloppc.p  |-  P  =  (oppCat `  D )
fulloppc.f  |-  ( ph  ->  F ( C Full  D
) G )
Assertion
Ref Expression
fulloppc  |-  ( ph  ->  F ( O Full  P
)tpos  G )

Proof of Theorem fulloppc
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fulloppc.o . . 3  |-  O  =  (oppCat `  C )
2 fulloppc.p . . 3  |-  P  =  (oppCat `  D )
3 fulloppc.f . . . 4  |-  ( ph  ->  F ( C Full  D
) G )
4 fullfunc 14104 . . . . 5  |-  ( C Full 
D )  C_  ( C  Func  D )
54ssbri 4255 . . . 4  |-  ( F ( C Full  D ) G  ->  F ( C  Func  D ) G )
63, 5syl 16 . . 3  |-  ( ph  ->  F ( C  Func  D ) G )
71, 2, 6funcoppc 14073 . 2  |-  ( ph  ->  F ( O  Func  P )tpos  G )
8 eqid 2437 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
9 eqid 2437 . . . . . 6  |-  (  Hom  `  D )  =  (  Hom  `  D )
10 eqid 2437 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
113adantr 453 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  F
( C Full  D ) G )
12 simprr 735 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
13 simprl 734 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
148, 9, 10, 11, 12, 13fullfo 14110 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
y G x ) : ( y (  Hom  `  C )
x ) -onto-> ( ( F `  y ) (  Hom  `  D
) ( F `  x ) ) )
15 forn 5657 . . . . 5  |-  ( ( y G x ) : ( y (  Hom  `  C )
x ) -onto-> ( ( F `  y ) (  Hom  `  D
) ( F `  x ) )  ->  ran  ( y G x )  =  ( ( F `  y ) (  Hom  `  D
) ( F `  x ) ) )
1614, 15syl 16 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ran  ( y G x )  =  ( ( F `  y ) (  Hom  `  D
) ( F `  x ) ) )
17 ovtpos 6495 . . . . 5  |-  ( xtpos 
G y )  =  ( y G x )
1817rneqi 5097 . . . 4  |-  ran  (
xtpos  G y )  =  ran  ( y G x )
199, 2oppchom 13942 . . . 4  |-  ( ( F `  x ) (  Hom  `  P
) ( F `  y ) )  =  ( ( F `  y ) (  Hom  `  D ) ( F `
 x ) )
2016, 18, 193eqtr4g 2494 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ran  ( xtpos  G y
)  =  ( ( F `  x ) (  Hom  `  P
) ( F `  y ) ) )
2120ralrimivva 2799 . 2  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) ran  ( xtpos  G y )  =  ( ( F `  x ) (  Hom  `  P
) ( F `  y ) ) )
221, 8oppcbas 13945 . . 3  |-  ( Base `  C )  =  (
Base `  O )
23 eqid 2437 . . 3  |-  (  Hom  `  P )  =  (  Hom  `  P )
2422, 23isfull 14108 . 2  |-  ( F ( O Full  P )tpos 
G  <->  ( F ( O  Func  P )tpos  G  /\  A. x  e.  ( Base `  C
) A. y  e.  ( Base `  C
) ran  ( xtpos  G y )  =  ( ( F `  x
) (  Hom  `  P
) ( F `  y ) ) ) )
257, 21, 24sylanbrc 647 1  |-  ( ph  ->  F ( O Full  P
)tpos  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706   class class class wbr 4213   ran crn 4880   -onto->wfo 5453   ` cfv 5455  (class class class)co 6082  tpos ctpos 6479   Basecbs 13470    Hom chom 13541  oppCatcoppc 13938    Func cfunc 14052   Full cful 14100
This theorem is referenced by:  ffthoppc  14122
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-tpos 6480  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-map 7021  df-ixp 7065  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-10 10067  df-n0 10223  df-z 10284  df-dec 10384  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-hom 13554  df-cco 13555  df-cat 13894  df-cid 13895  df-oppc 13939  df-func 14056  df-full 14102
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