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Theorem ordtcnv 17189
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 2389 . . . . . . . 8  |-  dom  R  =  dom  R
21psrn 14570 . . . . . . 7  |-  ( R  e.  PosetRel  ->  dom  R  =  ran  R )
32eqcomd 2394 . . . . . 6  |-  ( R  e.  PosetRel  ->  ran  R  =  dom  R )
43sneqd 3772 . . . . 5  |-  ( R  e.  PosetRel  ->  { ran  R }  =  { dom  R } )
5 vex 2904 . . . . . . . . . . . . 13  |-  y  e. 
_V
6 vex 2904 . . . . . . . . . . . . 13  |-  x  e. 
_V
75, 6brcnv 4997 . . . . . . . . . . . 12  |-  ( y `' R x  <->  x R
y )
87a1i 11 . . . . . . . . . . 11  |-  ( R  e.  PosetRel  ->  ( y `' R x  <->  x R
y ) )
98notbid 286 . . . . . . . . . 10  |-  ( R  e.  PosetRel  ->  ( -.  y `' R x  <->  -.  x R y ) )
103, 9rabeqbidv 2896 . . . . . . . . 9  |-  ( R  e.  PosetRel  ->  { y  e. 
ran  R  |  -.  y `' R x }  =  { y  e.  dom  R  |  -.  x R y } )
113, 10mpteq12dv 4230 . . . . . . . 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 5039 . . . . . . 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 4997 . . . . . . . . . . . 12  |-  ( x `' R y  <->  y R x )
1413a1i 11 . . . . . . . . . . 11  |-  ( R  e.  PosetRel  ->  ( x `' R y  <->  y R x ) )
1514notbid 286 . . . . . . . . . 10  |-  ( R  e.  PosetRel  ->  ( -.  x `' R y  <->  -.  y R x ) )
163, 15rabeqbidv 2896 . . . . . . . . 9  |-  ( R  e.  PosetRel  ->  { y  e. 
ran  R  |  -.  x `' R y }  =  { y  e.  dom  R  |  -.  y R x } )
173, 16mpteq12dv 4230 . . . . . . . 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 5039 . . . . . . 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 3447 . . . . . 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 3436 . . . . . 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 2437 . . . . 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 3447 . . . 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 5674 . . 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 5674 . 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 14573 . . 3  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
26 df-rn 4831 . . . 4  |-  ran  R  =  dom  `' R
27 eqid 2389 . . . 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 2389 . . . 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 17177 . . 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 16 . 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 2389 . . 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 2389 . . 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 17177 . 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 2431 1  |-  ( R  e.  PosetRel  ->  (ordTop `  `' R
)  =  (ordTop `  R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   {crab 2655    u. cun 3263   {csn 3759   class class class wbr 4155    e. cmpt 4209   `'ccnv 4819   dom cdm 4820   ran crn 4821   ` cfv 5396   ficfi 7352   topGenctg 13594  ordTopcordt 13650   PosetRelcps 14553
This theorem is referenced by:  ordtrest2  17192  cnvordtrestixx  24117
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-iota 5360  df-fun 5398  df-fv 5404  df-ordt 13654  df-ps 14558
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