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Theorem fulloppc 13812
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 13796 . . . . 5  |-  ( C Full 
D )  C_  ( C  Func  D )
54ssbri 4081 . . . 4  |-  ( F ( C Full  D ) G  ->  F ( C  Func  D ) G )
63, 5syl 15 . . 3  |-  ( ph  ->  F ( C  Func  D ) G )
71, 2, 6funcoppc 13765 . 2  |-  ( ph  ->  F ( O  Func  P )tpos  G )
8 eqid 2296 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
9 eqid 2296 . . . . . 6  |-  (  Hom  `  D )  =  (  Hom  `  D )
10 eqid 2296 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
113adantr 451 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  F
( C Full  D ) G )
12 simprr 733 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
13 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
148, 9, 10, 11, 12, 13fullfo 13802 . . . . 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 5470 . . . . 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 15 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ran  ( y G x )  =  ( ( F `  y ) (  Hom  `  D
) ( F `  x ) ) )
17 ovtpos 6265 . . . . 5  |-  ( xtpos 
G y )  =  ( y G x )
1817rneqi 4921 . . . 4  |-  ran  (
xtpos  G y )  =  ran  ( y G x )
199, 2oppchom 13634 . . . 4  |-  ( ( F `  x ) (  Hom  `  P
) ( F `  y ) )  =  ( ( F `  y ) (  Hom  `  D ) ( F `
 x ) )
2016, 18, 193eqtr4g 2353 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ran  ( xtpos  G y
)  =  ( ( F `  x ) (  Hom  `  P
) ( F `  y ) ) )
2120ralrimivva 2648 . 2  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) ran  ( xtpos  G y )  =  ( ( F `  x ) (  Hom  `  P
) ( F `  y ) ) )
221, 8oppcbas 13637 . . 3  |-  ( Base `  C )  =  (
Base `  O )
23 eqid 2296 . . 3  |-  (  Hom  `  P )  =  (  Hom  `  P )
2422, 23isfull 13800 . 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 645 1  |-  ( ph  ->  F ( O Full  P
)tpos  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039   ran crn 4706   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874  tpos ctpos 6249   Basecbs 13164    Hom chom 13235  oppCatcoppc 13630    Func cfunc 13744   Full cful 13792
This theorem is referenced by:  ffthoppc  13814
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-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-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  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-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-hom 13248  df-cco 13249  df-cat 13586  df-cid 13587  df-oppc 13631  df-func 13748  df-full 13794
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