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Theorem ordtcnv 16931
Description: The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
ordtcnv  |-  ( R  e.  PosetRel  ->  (ordTop `  `' R
)  =  (ordTop `  R ) )

Proof of Theorem ordtcnv
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . . . . . 8  |-  dom  R  =  dom  R
21psrn 14318 . . . . . . 7  |-  ( R  e.  PosetRel  ->  dom  R  =  ran  R )
32eqcomd 2288 . . . . . 6  |-  ( R  e.  PosetRel  ->  ran  R  =  dom  R )
43sneqd 3653 . . . . 5  |-  ( R  e.  PosetRel  ->  { ran  R }  =  { dom  R } )
5 vex 2791 . . . . . . . . . . . . 13  |-  y  e. 
_V
6 vex 2791 . . . . . . . . . . . . 13  |-  x  e. 
_V
75, 6brcnv 4864 . . . . . . . . . . . 12  |-  ( y `' R x  <->  x R
y )
87a1i 10 . . . . . . . . . . 11  |-  ( R  e.  PosetRel  ->  ( y `' R x  <->  x R
y ) )
98notbid 285 . . . . . . . . . 10  |-  ( R  e.  PosetRel  ->  ( -.  y `' R x  <->  -.  x R y ) )
103, 9rabeqbidv 2783 . . . . . . . . 9  |-  ( R  e.  PosetRel  ->  { y  e. 
ran  R  |  -.  y `' R x }  =  { y  e.  dom  R  |  -.  x R y } )
113, 10mpteq12dv 4098 . . . . . . . 8  |-  ( R  e.  PosetRel  ->  ( x  e. 
ran  R  |->  { y  e.  ran  R  |  -.  y `' R x } )  =  ( x  e.  dom  R  |->  { y  e.  dom  R  |  -.  x R y } ) )
1211rneqd 4906 . . . . . . 7  |-  ( R  e.  PosetRel  ->  ran  ( x  e.  ran  R  |->  { y  e.  ran  R  |  -.  y `' R x } )  =  ran  ( x  e.  dom  R 
|->  { y  e.  dom  R  |  -.  x R y } ) )
136, 5brcnv 4864 . . . . . . . . . . . 12  |-  ( x `' R y  <->  y R x )
1413a1i 10 . . . . . . . . . . 11  |-  ( R  e.  PosetRel  ->  ( x `' R y  <->  y R x ) )
1514notbid 285 . . . . . . . . . 10  |-  ( R  e.  PosetRel  ->  ( -.  x `' R y  <->  -.  y R x ) )
163, 15rabeqbidv 2783 . . . . . . . . 9  |-  ( R  e.  PosetRel  ->  { y  e. 
ran  R  |  -.  x `' R y }  =  { y  e.  dom  R  |  -.  y R x } )
173, 16mpteq12dv 4098 . . . . . . . 8  |-  ( R  e.  PosetRel  ->  ( x  e. 
ran  R  |->  { y  e.  ran  R  |  -.  x `' R y } )  =  ( x  e.  dom  R  |->  { y  e.  dom  R  |  -.  y R x } ) )
1817rneqd 4906 . . . . . . 7  |-  ( R  e.  PosetRel  ->  ran  ( x  e.  ran  R  |->  { y  e.  ran  R  |  -.  x `' R y } )  =  ran  ( x  e.  dom  R 
|->  { y  e.  dom  R  |  -.  y R x } ) )
1912, 18uneq12d 3330 . . . . . 6  |-  ( R  e.  PosetRel  ->  ( ran  (
x  e.  ran  R  |->  { y  e.  ran  R  |  -.  y `' R x } )  u.  ran  ( x  e.  ran  R  |->  { y  e.  ran  R  |  -.  x `' R
y } ) )  =  ( ran  (
x  e.  dom  R  |->  { y  e.  dom  R  |  -.  x R y } )  u. 
ran  ( x  e. 
dom  R  |->  { y  e.  dom  R  |  -.  y R x }
) ) )
20 uncom 3319 . . . . . 6  |-  ( ran  ( x  e.  dom  R 
|->  { y  e.  dom  R  |  -.  x R y } )  u. 
ran  ( x  e. 
dom  R  |->  { y  e.  dom  R  |  -.  y R x }
) )  =  ( ran  ( x  e. 
dom  R  |->  { y  e.  dom  R  |  -.  y R x }
)  u.  ran  (
x  e.  dom  R  |->  { y  e.  dom  R  |  -.  x R y } ) )
2119, 20syl6eq 2331 . . . . 5  |-  ( R  e.  PosetRel  ->  ( ran  (
x  e.  ran  R  |->  { y  e.  ran  R  |  -.  y `' R x } )  u.  ran  ( x  e.  ran  R  |->  { y  e.  ran  R  |  -.  x `' R
y } ) )  =  ( ran  (
x  e.  dom  R  |->  { y  e.  dom  R  |  -.  y R x } )  u. 
ran  ( x  e. 
dom  R  |->  { y  e.  dom  R  |  -.  x R y } ) ) )
224, 21uneq12d 3330 . . . 4  |-  ( R  e.  PosetRel  ->  ( { ran  R }  u.  ( ran  ( x  e.  ran  R 
|->  { y  e.  ran  R  |  -.  y `' R x } )  u.  ran  ( x  e.  ran  R  |->  { y  e.  ran  R  |  -.  x `' R
y } ) ) )  =  ( { dom  R }  u.  ( ran  ( x  e. 
dom  R  |->  { y  e.  dom  R  |  -.  y R x }
)  u.  ran  (
x  e.  dom  R  |->  { y  e.  dom  R  |  -.  x R y } ) ) ) )
2322fveq2d 5529 . . 3  |-  ( R  e.  PosetRel  ->  ( fi `  ( { ran  R }  u.  ( ran  ( x  e.  ran  R  |->  { y  e.  ran  R  |  -.  y `' R x } )  u.  ran  ( x  e.  ran  R 
|->  { y  e.  ran  R  |  -.  x `' R y } ) ) ) )  =  ( fi `  ( { dom  R }  u.  ( ran  ( x  e. 
dom  R  |->  { y  e.  dom  R  |  -.  y R x }
)  u.  ran  (
x  e.  dom  R  |->  { y  e.  dom  R  |  -.  x R y } ) ) ) ) )
2423fveq2d 5529 . 2  |-  ( R  e.  PosetRel  ->  ( topGen `  ( fi `  ( { ran  R }  u.  ( ran  ( x  e.  ran  R 
|->  { y  e.  ran  R  |  -.  y `' R x } )  u.  ran  ( x  e.  ran  R  |->  { y  e.  ran  R  |  -.  x `' R
y } ) ) ) ) )  =  ( topGen `  ( fi `  ( { dom  R }  u.  ( ran  ( x  e.  dom  R 
|->  { y  e.  dom  R  |  -.  y R x } )  u. 
ran  ( x  e. 
dom  R  |->  { y  e.  dom  R  |  -.  x R y } ) ) ) ) ) )
25 cnvps 14321 . . 3  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
26 df-rn 4700 . . . 4  |-  ran  R  =  dom  `' R
27 eqid 2283 . . . 4  |-  ran  (
x  e.  ran  R  |->  { y  e.  ran  R  |  -.  y `' R x } )  =  ran  ( x  e.  ran  R  |->  { y  e.  ran  R  |  -.  y `' R x } )
28 eqid 2283 . . . 4  |-  ran  (
x  e.  ran  R  |->  { y  e.  ran  R  |  -.  x `' R y } )  =  ran  ( x  e.  ran  R  |->  { y  e.  ran  R  |  -.  x `' R
y } )
2926, 27, 28ordtval 16919 . . 3  |-  ( `' R  e.  PosetRel  ->  (ordTop `  `' R )  =  (
topGen `  ( fi `  ( { ran  R }  u.  ( ran  ( x  e.  ran  R  |->  { y  e.  ran  R  |  -.  y `' R x } )  u.  ran  ( x  e.  ran  R 
|->  { y  e.  ran  R  |  -.  x `' R y } ) ) ) ) ) )
3025, 29syl 15 . 2  |-  ( R  e.  PosetRel  ->  (ordTop `  `' R
)  =  ( topGen `  ( fi `  ( { ran  R }  u.  ( ran  ( x  e. 
ran  R  |->  { y  e.  ran  R  |  -.  y `' R x } )  u.  ran  ( x  e.  ran  R 
|->  { y  e.  ran  R  |  -.  x `' R y } ) ) ) ) ) )
31 eqid 2283 . . 3  |-  ran  (
x  e.  dom  R  |->  { y  e.  dom  R  |  -.  y R x } )  =  ran  ( x  e. 
dom  R  |->  { y  e.  dom  R  |  -.  y R x }
)
32 eqid 2283 . . 3  |-  ran  (
x  e.  dom  R  |->  { y  e.  dom  R  |  -.  x R y } )  =  ran  ( x  e. 
dom  R  |->  { y  e.  dom  R  |  -.  x R y } )
331, 31, 32ordtval 16919 . 2  |-  ( R  e.  PosetRel  ->  (ordTop `  R )  =  ( topGen `  ( fi `  ( { dom  R }  u.  ( ran  ( x  e.  dom  R 
|->  { y  e.  dom  R  |  -.  y R x } )  u. 
ran  ( x  e. 
dom  R  |->  { y  e.  dom  R  |  -.  x R y } ) ) ) ) ) )
3424, 30, 333eqtr4d 2325 1  |-  ( R  e.  PosetRel  ->  (ordTop `  `' R
)  =  (ordTop `  R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   {crab 2547    u. cun 3150   {csn 3640   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   ran crn 4690   ` cfv 5255   ficfi 7164   topGenctg 13342  ordTopcordt 13398   PosetRelcps 14301
This theorem is referenced by:  ordtrest2  16934  cnvordtrestixx  23297
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-iota 5219  df-fun 5257  df-fv 5263  df-ordt 13402  df-ps 14306
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