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Theorem dualalg 25782
Description: The dual of a  Alg is a  Alg. (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
dualalg  |-  ( T  e.  Alg  ->  <. <. C ,  D >. ,  <. J , tpos  R
>. >.  e.  Alg  )

Proof of Theorem dualalg
StepHypRef Expression
1 eqid 2283 . . . . 5  |-  dom  D  =  dom  D
2 dualcat2.1 . . . . 5  |-  D  =  ( dom_ `  T
)
3 eqid 2283 . . . . 5  |-  dom  J  =  dom  J
4 dualcat2.3 . . . . 5  |-  J  =  ( id_ `  T
)
5 dualcat2.2 . . . . 5  |-  C  =  ( cod_ `  T
)
61, 2, 3, 4, 5coda 25729 . . . 4  |-  ( T  e.  Alg  ->  C : dom  D --> dom  J
)
72, 5dcsda 25733 . . . . 5  |-  ( T  e.  Alg  ->  dom  D  =  dom  C )
87feq2d 5380 . . . 4  |-  ( T  e.  Alg  ->  ( C : dom  D --> dom  J  <->  C : dom  C --> dom  J
) )
96, 8mpbid 201 . . 3  |-  ( T  e.  Alg  ->  C : dom  C --> dom  J
)
101, 2, 3, 4doma 25728 . . . 4  |-  ( T  e.  Alg  ->  D : dom  D --> dom  J
)
117feq2d 5380 . . . 4  |-  ( T  e.  Alg  ->  ( D : dom  D --> dom  J  <->  D : dom  C --> dom  J
) )
1210, 11mpbid 201 . . 3  |-  ( T  e.  Alg  ->  D : dom  C --> dom  J
)
131, 2, 3, 4ida 25730 . . . 4  |-  ( T  e.  Alg  ->  J : dom  J --> dom  D
)
14 feq3 5377 . . . . 5  |-  ( dom 
D  =  dom  C  ->  ( J : dom  J --> dom  D  <->  J : dom  J --> dom  C )
)
157, 14syl 15 . . . 4  |-  ( T  e.  Alg  ->  ( J : dom  J --> dom  D  <->  J : dom  J --> dom  C
) )
1613, 15mpbid 201 . . 3  |-  ( T  e.  Alg  ->  J : dom  J --> dom  C
)
179, 12, 163jca 1132 . 2  |-  ( T  e.  Alg  ->  ( C : dom  C --> dom  J  /\  D : dom  C --> dom  J  /\  J : dom  J --> dom  C )
)
18 dualcat2.4 . . . . . 6  |-  R  =  ( o_ `  T
)
191, 2, 18cmppfa 25732 . . . . 5  |-  ( T  e.  Alg  ->  ( Fun  R  /\  dom  R  C_  ( dom  D  X.  dom  D )  /\  ran  R 
C_  dom  D )
)
2019simp1d 967 . . . 4  |-  ( T  e.  Alg  ->  Fun  R )
21 tposfun 6250 . . . 4  |-  ( Fun 
R  ->  Fun tpos  R )
2220, 21syl 15 . . 3  |-  ( T  e.  Alg  ->  Fun tpos  R )
2319simp2d 968 . . . . 5  |-  ( T  e.  Alg  ->  dom  R 
C_  ( dom  D  X.  dom  D ) )
24 cnvss 4854 . . . . 5  |-  ( dom 
R  C_  ( dom  D  X.  dom  D )  ->  `' dom  R  C_  `' ( dom  D  X.  dom  D ) )
2523, 24syl 15 . . . 4  |-  ( T  e.  Alg  ->  `' dom  R  C_  `' ( dom  D  X.  dom  D
) )
26 relxp 4794 . . . . . 6  |-  Rel  ( dom  D  X.  dom  D
)
27 relss 4775 . . . . . 6  |-  ( dom 
R  C_  ( dom  D  X.  dom  D )  ->  ( Rel  ( dom  D  X.  dom  D
)  ->  Rel  dom  R
) )
2823, 26, 27ee10 1366 . . . . 5  |-  ( T  e.  Alg  ->  Rel  dom 
R )
29 dmtpos 6246 . . . . 5  |-  ( Rel 
dom  R  ->  dom tpos  R  =  `' dom  R )
3028, 29syl 15 . . . 4  |-  ( T  e.  Alg  ->  dom tpos  R  =  `' dom  R
)
31 cnvxp 5097 . . . . 5  |-  `' ( dom  D  X.  dom  D )  =  ( dom 
D  X.  dom  D
)
327, 7xpeq12d 4714 . . . . 5  |-  ( T  e.  Alg  ->  ( dom  D  X.  dom  D
)  =  ( dom 
C  X.  dom  C
) )
3331, 32syl5req 2328 . . . 4  |-  ( T  e.  Alg  ->  ( dom  C  X.  dom  C
)  =  `' ( dom  D  X.  dom  D ) )
3425, 30, 333sstr4d 3221 . . 3  |-  ( T  e.  Alg  ->  dom tpos  R 
C_  ( dom  C  X.  dom  C ) )
35 rntpos 6247 . . . . 5  |-  ( Rel 
dom  R  ->  ran tpos  R  =  ran  R )
3628, 35syl 15 . . . 4  |-  ( T  e.  Alg  ->  ran tpos  R  =  ran  R )
3719simp3d 969 . . . . 5  |-  ( T  e.  Alg  ->  ran  R 
C_  dom  D )
3837, 7sseqtrd 3214 . . . 4  |-  ( T  e.  Alg  ->  ran  R 
C_  dom  C )
3936, 38eqsstrd 3212 . . 3  |-  ( T  e.  Alg  ->  ran tpos  R 
C_  dom  C )
4022, 34, 393jca 1132 . 2  |-  ( T  e.  Alg  ->  ( Fun tpos  R  /\  dom tpos  R  C_  ( dom  C  X.  dom  C )  /\  ran tpos  R  C_  dom  C ) )
41 fvex 5539 . . . . . 6  |-  ( cod_ `  T )  e.  _V
425, 41eqeltri 2353 . . . . 5  |-  C  e. 
_V
43 fvex 5539 . . . . . 6  |-  ( dom_ `  T )  e.  _V
442, 43eqeltri 2353 . . . . 5  |-  D  e. 
_V
45 fvex 5539 . . . . . 6  |-  ( id_ `  T )  e.  _V
464, 45eqeltri 2353 . . . . 5  |-  J  e. 
_V
4742, 44, 463pm3.2i 1130 . . . 4  |-  ( C  e.  _V  /\  D  e.  _V  /\  J  e. 
_V )
48 fvex 5539 . . . . . 6  |-  ( o_
`  T )  e. 
_V
4918, 48eqeltri 2353 . . . . 5  |-  R  e. 
_V
5049tposex 6268 . . . 4  |- tpos  R  e. 
_V
5147, 50pm3.2i 441 . . 3  |-  ( ( C  e.  _V  /\  D  e.  _V  /\  J  e.  _V )  /\ tpos  R  e. 
_V )
52 eqid 2283 . . . 4  |-  dom  C  =  dom  C
5352, 3isalg 25721 . . 3  |-  ( ( ( C  e.  _V  /\  D  e.  _V  /\  J  e.  _V )  /\ tpos  R  e.  _V )  ->  ( <. <. C ,  D >. ,  <. J , tpos  R >. >.  e.  Alg  <->  ( ( C : dom  C --> dom  J  /\  D : dom  C --> dom  J  /\  J : dom  J --> dom  C )  /\  ( Fun tpos  R  /\  dom tpos  R  C_  ( dom  C  X.  dom  C )  /\  ran tpos  R  C_  dom  C ) ) ) )
5451, 53mp1i 11 . 2  |-  ( T  e.  Alg  ->  ( <. <. C ,  D >. ,  <. J , tpos  R >. >.  e.  Alg  <->  ( ( C : dom  C --> dom  J  /\  D : dom  C --> dom  J  /\  J : dom  J --> dom  C )  /\  ( Fun tpos  R  /\  dom tpos  R  C_  ( dom  C  X.  dom  C )  /\  ran tpos  R  C_  dom  C ) ) ) )
5517, 40, 54mpbir2and 888 1  |-  ( T  e.  Alg  ->  <. <. C ,  D >. ,  <. J , tpos  R
>. >.  e.  Alg  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   <.cop 3643    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   Rel wrel 4694   Fun wfun 5249   -->wf 5251   ` cfv 5255  tpos ctpos 6233    Alg calg 25711   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714   o_co_ 25715
This theorem is referenced by:  dualded  25783
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-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-1st 6122  df-2nd 6123  df-tpos 6234  df-alg 25716  df-dom_ 25717  df-cod_ 25718  df-id_ 25719  df-cmpa 25720
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