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Theorem dualded 25783
Description: The dual of a deductive system is a deductive system. (Contributed by FL, 18-Apr-2010.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
Hypotheses
Ref Expression
dualcat2.1  |-  D  =  ( dom_ `  T
)
dualcat2.2  |-  C  =  ( cod_ `  T
)
dualcat2.3  |-  J  =  ( id_ `  T
)
dualcat2.4  |-  R  =  ( o_ `  T
)
Assertion
Ref Expression
dualded  |-  ( T  e.  Ded  ->  <. <. C ,  D >. ,  <. J , tpos  R
>. >.  e.  Ded )

Proof of Theorem dualded
Dummy variables  a 
f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dedalg 25743 . . . 4  |-  ( T  e.  Ded  ->  T  e.  Alg  )
2 dualcat2.1 . . . . 5  |-  D  =  ( dom_ `  T
)
3 dualcat2.2 . . . . 5  |-  C  =  ( cod_ `  T
)
4 dualcat2.3 . . . . 5  |-  J  =  ( id_ `  T
)
5 dualcat2.4 . . . . 5  |-  R  =  ( o_ `  T
)
62, 3, 4, 5dualalg 25782 . . . 4  |-  ( T  e.  Alg  ->  <. <. C ,  D >. ,  <. J , tpos  R
>. >.  e.  Alg  )
71, 6syl 15 . . 3  |-  ( T  e.  Ded  ->  <. <. C ,  D >. ,  <. J , tpos  R
>. >.  e.  Alg  )
8 eqid 2283 . . . . . 6  |-  dom  J  =  dom  J
98, 2, 4, 3idosd 25744 . . . . 5  |-  ( ( T  e.  Ded  /\  a  e.  dom  J )  ->  ( ( D `
 ( J `  a ) )  =  a  /\  ( C `
 ( J `  a ) )  =  a ) )
109ancomd 438 . . . 4  |-  ( ( T  e.  Ded  /\  a  e.  dom  J )  ->  ( ( C `
 ( J `  a ) )  =  a  /\  ( D `
 ( J `  a ) )  =  a ) )
1110ralrimiva 2626 . . 3  |-  ( T  e.  Ded  ->  A. a  e.  dom  J ( ( C `  ( J `
 a ) )  =  a  /\  ( D `  ( J `  a ) )  =  a ) )
12 ancom 437 . . . . . 6  |-  ( ( g  e.  dom  D  /\  f  e.  dom  D )  <->  ( f  e. 
dom  D  /\  g  e.  dom  D ) )
132, 3dcsda 25733 . . . . . . . . 9  |-  ( T  e.  Alg  ->  dom  D  =  dom  C )
141, 13syl 15 . . . . . . . 8  |-  ( T  e.  Ded  ->  dom  D  =  dom  C )
1514eleq2d 2350 . . . . . . 7  |-  ( T  e.  Ded  ->  (
f  e.  dom  D  <->  f  e.  dom  C ) )
1614eleq2d 2350 . . . . . . 7  |-  ( T  e.  Ded  ->  (
g  e.  dom  D  <->  g  e.  dom  C ) )
1715, 16anbi12d 691 . . . . . 6  |-  ( T  e.  Ded  ->  (
( f  e.  dom  D  /\  g  e.  dom  D )  <->  ( f  e. 
dom  C  /\  g  e.  dom  C ) ) )
1812, 17syl5rbb 249 . . . . 5  |-  ( T  e.  Ded  ->  (
( f  e.  dom  C  /\  g  e.  dom  C )  <->  ( g  e. 
dom  D  /\  f  e.  dom  D ) ) )
19 eqid 2283 . . . . . . . . . . . . . 14  |-  dom  D  =  dom  D
2019, 2, 5cmppfa 25732 . . . . . . . . . . . . 13  |-  ( T  e.  Alg  ->  ( Fun  R  /\  dom  R  C_  ( dom  D  X.  dom  D )  /\  ran  R 
C_  dom  D )
)
211, 20syl 15 . . . . . . . . . . . 12  |-  ( T  e.  Ded  ->  ( Fun  R  /\  dom  R  C_  ( dom  D  X.  dom  D )  /\  ran  R 
C_  dom  D )
)
2221simp2d 968 . . . . . . . . . . 11  |-  ( T  e.  Ded  ->  dom  R 
C_  ( dom  D  X.  dom  D ) )
23 relxp 4794 . . . . . . . . . . 11  |-  Rel  ( dom  D  X.  dom  D
)
24 relss 4775 . . . . . . . . . . 11  |-  ( dom 
R  C_  ( dom  D  X.  dom  D )  ->  ( Rel  ( dom  D  X.  dom  D
)  ->  Rel  dom  R
) )
2522, 23, 24ee10 1366 . . . . . . . . . 10  |-  ( T  e.  Ded  ->  Rel  dom 
R )
26 dmtpos 6246 . . . . . . . . . 10  |-  ( Rel 
dom  R  ->  dom tpos  R  =  `' dom  R )
2725, 26syl 15 . . . . . . . . 9  |-  ( T  e.  Ded  ->  dom tpos  R  =  `' dom  R
)
28273ad2ant1 976 . . . . . . . 8  |-  ( ( T  e.  Ded  /\  g  e.  dom  D  /\  f  e.  dom  D )  ->  dom tpos  R  =  `' dom  R )
2928eleq2d 2350 . . . . . . 7  |-  ( ( T  e.  Ded  /\  g  e.  dom  D  /\  f  e.  dom  D )  ->  ( <. g ,  f >.  e.  dom tpos  R  <->  <. g ,  f >.  e.  `' dom  R ) )
3019, 2, 3, 5cmppfd 25745 . . . . . . . 8  |-  ( ( T  e.  Ded  /\  g  e.  dom  D  /\  f  e.  dom  D )  ->  ( <. f ,  g >.  e.  dom  R  <-> 
( D `  f
)  =  ( C `
 g ) ) )
31 vex 2791 . . . . . . . . 9  |-  g  e. 
_V
32 vex 2791 . . . . . . . . 9  |-  f  e. 
_V
3331, 32opelcnv 4863 . . . . . . . 8  |-  ( <.
g ,  f >.  e.  `' dom  R  <->  <. f ,  g >.  e.  dom  R )
34 eqcom 2285 . . . . . . . 8  |-  ( ( C `  g )  =  ( D `  f )  <->  ( D `  f )  =  ( C `  g ) )
3530, 33, 343bitr4g 279 . . . . . . 7  |-  ( ( T  e.  Ded  /\  g  e.  dom  D  /\  f  e.  dom  D )  ->  ( <. g ,  f >.  e.  `' dom  R  <->  ( C `  g )  =  ( D `  f ) ) )
3629, 35bitrd 244 . . . . . 6  |-  ( ( T  e.  Ded  /\  g  e.  dom  D  /\  f  e.  dom  D )  ->  ( <. g ,  f >.  e.  dom tpos  R  <-> 
( C `  g
)  =  ( D `
 f ) ) )
37363expib 1154 . . . . 5  |-  ( T  e.  Ded  ->  (
( g  e.  dom  D  /\  f  e.  dom  D )  ->  ( <. g ,  f >.  e.  dom tpos  R  <-> 
( C `  g
)  =  ( D `
 f ) ) ) )
3818, 37sylbid 206 . . . 4  |-  ( T  e.  Ded  ->  (
( f  e.  dom  C  /\  g  e.  dom  C )  ->  ( <. g ,  f >.  e.  dom tpos  R  <-> 
( C `  g
)  =  ( D `
 f ) ) ) )
3938ralrimivv 2634 . . 3  |-  ( T  e.  Ded  ->  A. f  e.  dom  C A. g  e.  dom  C ( <.
g ,  f >.  e.  dom tpos  R  <->  ( C `  g )  =  ( D `  f ) ) )
407, 11, 393jca 1132 . 2  |-  ( T  e.  Ded  ->  ( <. <. C ,  D >. ,  <. J , tpos  R >. >.  e.  Alg  /\  A. a  e.  dom  J
( ( C `  ( J `  a ) )  =  a  /\  ( D `  ( J `
 a ) )  =  a )  /\  A. f  e.  dom  C A. g  e.  dom  C ( <. g ,  f
>.  e.  dom tpos  R  <->  ( C `  g )  =  ( D `  f ) ) ) )
4119, 2, 3, 5codcmpd 25747 . . . . . . 7  |-  ( ( T  e.  Ded  /\  g  e.  dom  D  /\  f  e.  dom  D )  ->  ( ( D `
 f )  =  ( C `  g
)  ->  ( C `  ( f R g ) )  =  ( C `  f ) ) )
42 ovtpos 6249 . . . . . . . . 9  |-  ( gtpos 
R f )  =  ( f R g )
4342fveq2i 5528 . . . . . . . 8  |-  ( C `
 ( gtpos  R
f ) )  =  ( C `  (
f R g ) )
4443eqeq1i 2290 . . . . . . 7  |-  ( ( C `  ( gtpos 
R f ) )  =  ( C `  f )  <->  ( C `  ( f R g ) )  =  ( C `  f ) )
4541, 34, 443imtr4g 261 . . . . . 6  |-  ( ( T  e.  Ded  /\  g  e.  dom  D  /\  f  e.  dom  D )  ->  ( ( C `
 g )  =  ( D `  f
)  ->  ( C `  ( gtpos  R f ) )  =  ( C `  f ) ) )
46453expib 1154 . . . . 5  |-  ( T  e.  Ded  ->  (
( g  e.  dom  D  /\  f  e.  dom  D )  ->  ( ( C `  g )  =  ( D `  f )  ->  ( C `  ( gtpos  R f ) )  =  ( C `  f ) ) ) )
4718, 46sylbid 206 . . . 4  |-  ( T  e.  Ded  ->  (
( f  e.  dom  C  /\  g  e.  dom  C )  ->  ( ( C `  g )  =  ( D `  f )  ->  ( C `  ( gtpos  R f ) )  =  ( C `  f ) ) ) )
4847ralrimivv 2634 . . 3  |-  ( T  e.  Ded  ->  A. f  e.  dom  C A. g  e.  dom  C ( ( C `  g )  =  ( D `  f )  ->  ( C `  ( gtpos  R f ) )  =  ( C `  f ) ) )
4919, 2, 3, 5domcmpd 25746 . . . . . . 7  |-  ( ( T  e.  Ded  /\  g  e.  dom  D  /\  f  e.  dom  D )  ->  ( ( D `
 f )  =  ( C `  g
)  ->  ( D `  ( f R g ) )  =  ( D `  g ) ) )
5042fveq2i 5528 . . . . . . . 8  |-  ( D `
 ( gtpos  R
f ) )  =  ( D `  (
f R g ) )
5150eqeq1i 2290 . . . . . . 7  |-  ( ( D `  ( gtpos 
R f ) )  =  ( D `  g )  <->  ( D `  ( f R g ) )  =  ( D `  g ) )
5249, 34, 513imtr4g 261 . . . . . 6  |-  ( ( T  e.  Ded  /\  g  e.  dom  D  /\  f  e.  dom  D )  ->  ( ( C `
 g )  =  ( D `  f
)  ->  ( D `  ( gtpos  R f ) )  =  ( D `  g ) ) )
53523expib 1154 . . . . 5  |-  ( T  e.  Ded  ->  (
( g  e.  dom  D  /\  f  e.  dom  D )  ->  ( ( C `  g )  =  ( D `  f )  ->  ( D `  ( gtpos  R f ) )  =  ( D `  g ) ) ) )
5418, 53sylbid 206 . . . 4  |-  ( T  e.  Ded  ->  (
( f  e.  dom  C  /\  g  e.  dom  C )  ->  ( ( C `  g )  =  ( D `  f )  ->  ( D `  ( gtpos  R f ) )  =  ( D `  g ) ) ) )
5554ralrimivv 2634 . . 3  |-  ( T  e.  Ded  ->  A. f  e.  dom  C A. g  e.  dom  C ( ( C `  g )  =  ( D `  f )  ->  ( D `  ( gtpos  R f ) )  =  ( D `  g ) ) )
5648, 55jca 518 . 2  |-  ( T  e.  Ded  ->  ( A. f  e.  dom  C A. g  e.  dom  C ( ( C `  g )  =  ( D `  f )  ->  ( C `  ( gtpos  R f
) )  =  ( C `  f ) )  /\  A. f  e.  dom  C A. g  e.  dom  C ( ( C `  g )  =  ( D `  f )  ->  ( D `  ( gtpos  R f ) )  =  ( D `  g ) ) ) )
57 fvex 5539 . . . 4  |-  ( cod_ `  T )  e.  _V
583, 57eqeltri 2353 . . 3  |-  C  e. 
_V
59 fvex 5539 . . . 4  |-  ( dom_ `  T )  e.  _V
602, 59eqeltri 2353 . . 3  |-  D  e. 
_V
61 fvex 5539 . . . 4  |-  ( id_ `  T )  e.  _V
624, 61eqeltri 2353 . . 3  |-  J  e. 
_V
63 fvex 5539 . . . . . 6  |-  ( o_
`  T )  e. 
_V
645, 63eqeltri 2353 . . . . 5  |-  R  e. 
_V
6564tposex 6268 . . . 4  |- tpos  R  e. 
_V
66 eqid 2283 . . . . 5  |-  dom  C  =  dom  C
6766, 8isded 25736 . . . 4  |-  ( ( ( C  e.  _V  /\  D  e.  _V  /\  J  e.  _V )  /\ tpos  R  e.  _V )  ->  ( <. <. C ,  D >. ,  <. J , tpos  R >. >.  e.  Ded  <->  ( ( <. <. C ,  D >. ,  <. J , tpos  R >. >.  e.  Alg  /\  A. a  e.  dom  J
( ( C `  ( J `  a ) )  =  a  /\  ( D `  ( J `
 a ) )  =  a )  /\  A. f  e.  dom  C A. g  e.  dom  C ( <. g ,  f
>.  e.  dom tpos  R  <->  ( C `  g )  =  ( D `  f ) ) )  /\  ( A. f  e.  dom  C A. g  e.  dom  C ( ( C `  g )  =  ( D `  f )  ->  ( C `  ( gtpos  R f
) )  =  ( C `  f ) )  /\  A. f  e.  dom  C A. g  e.  dom  C ( ( C `  g )  =  ( D `  f )  ->  ( D `  ( gtpos  R f ) )  =  ( D `  g ) ) ) ) ) )
6865, 67mpan2 652 . . 3  |-  ( ( C  e.  _V  /\  D  e.  _V  /\  J  e.  _V )  ->  ( <. <. C ,  D >. ,  <. J , tpos  R >. >.  e.  Ded  <->  ( ( <. <. C ,  D >. ,  <. J , tpos  R >. >.  e.  Alg  /\  A. a  e.  dom  J
( ( C `  ( J `  a ) )  =  a  /\  ( D `  ( J `
 a ) )  =  a )  /\  A. f  e.  dom  C A. g  e.  dom  C ( <. g ,  f
>.  e.  dom tpos  R  <->  ( C `  g )  =  ( D `  f ) ) )  /\  ( A. f  e.  dom  C A. g  e.  dom  C ( ( C `  g )  =  ( D `  f )  ->  ( C `  ( gtpos  R f
) )  =  ( C `  f ) )  /\  A. f  e.  dom  C A. g  e.  dom  C ( ( C `  g )  =  ( D `  f )  ->  ( D `  ( gtpos  R f ) )  =  ( D `  g ) ) ) ) ) )
6958, 60, 62, 68mp3an 1277 . 2  |-  ( <. <. C ,  D >. , 
<. J , tpos  R >. >.  e.  Ded  <->  ( ( <. <. C ,  D >. , 
<. J , tpos  R >. >.  e.  Alg  /\  A. a  e.  dom  J ( ( C `  ( J `
 a ) )  =  a  /\  ( D `  ( J `  a ) )  =  a )  /\  A. f  e.  dom  C A. g  e.  dom  C (
<. g ,  f >.  e.  dom tpos  R  <->  ( C `  g )  =  ( D `  f ) ) )  /\  ( A. f  e.  dom  C A. g  e.  dom  C ( ( C `  g )  =  ( D `  f )  ->  ( C `  ( gtpos  R f
) )  =  ( C `  f ) )  /\  A. f  e.  dom  C A. g  e.  dom  C ( ( C `  g )  =  ( D `  f )  ->  ( D `  ( gtpos  R f ) )  =  ( D `  g ) ) ) ) )
7040, 56, 69sylanbrc 645 1  |-  ( T  e.  Ded  ->  <. <. C ,  D >. ,  <. J , tpos  R
>. >.  e.  Ded )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   <.cop 3643    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   Rel wrel 4694   Fun wfun 5249   ` cfv 5255  (class class class)co 5858  tpos ctpos 6233    Alg calg 25711   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714   o_co_ 25715   Dedcded 25734
This theorem is referenced by:  dualcat2  25784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-tpos 6234  df-alg 25716  df-dom_ 25717  df-cod_ 25718  df-id_ 25719  df-cmpa 25720  df-ded 25735
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