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Theorem fulloppc 15533
Description: The opposite functor of a full functor is also full. Proposition 3.43(d) in [Adamek] p. 39. (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 15517 . . . . 5  |-  ( C Full 
D )  C_  ( C  Func  D )
54ssbri 4436 . . . 4  |-  ( F ( C Full  D ) G  ->  F ( C  Func  D ) G )
63, 5syl 17 . . 3  |-  ( ph  ->  F ( C  Func  D ) G )
71, 2, 6funcoppc 15486 . 2  |-  ( ph  ->  F ( O  Func  P )tpos  G )
8 eqid 2402 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
9 eqid 2402 . . . . . 6  |-  ( Hom  `  D )  =  ( Hom  `  D )
10 eqid 2402 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
113adantr 463 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  F
( C Full  D ) G )
12 simprr 758 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
13 simprl 756 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
148, 9, 10, 11, 12, 13fullfo 15523 . . . . 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 5780 . . . . 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 17 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ran  ( y G x )  =  ( ( F `  y ) ( Hom  `  D
) ( F `  x ) ) )
17 ovtpos 6972 . . . . 5  |-  ( xtpos 
G y )  =  ( y G x )
1817rneqi 5049 . . . 4  |-  ran  (
xtpos  G y )  =  ran  ( y G x )
199, 2oppchom 15326 . . . 4  |-  ( ( F `  x ) ( Hom  `  P
) ( F `  y ) )  =  ( ( F `  y ) ( Hom  `  D ) ( F `
 x ) )
2016, 18, 193eqtr4g 2468 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  ran  ( xtpos  G y
)  =  ( ( F `  x ) ( Hom  `  P
) ( F `  y ) ) )
2120ralrimivva 2824 . 2  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) ran  ( xtpos  G y )  =  ( ( F `  x ) ( Hom  `  P
) ( F `  y ) ) )
221, 8oppcbas 15329 . . 3  |-  ( Base `  C )  =  (
Base `  O )
23 eqid 2402 . . 3  |-  ( Hom  `  P )  =  ( Hom  `  P )
2422, 23isfull 15521 . 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 662 1  |-  ( ph  ->  F ( O Full  P
)tpos  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   class class class wbr 4394   ran crn 4823   -onto->wfo 5566   ` cfv 5568  (class class class)co 6277  tpos ctpos 6956   Basecbs 14839   Hom chom 14918  oppCatcoppc 15322    Func cfunc 15465   Full cful 15513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-tpos 6957  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-map 7458  df-ixp 7507  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-hom 14931  df-cco 14932  df-cat 15280  df-cid 15281  df-oppc 15323  df-func 15469  df-full 15515
This theorem is referenced by:  ffthoppc  15535
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