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Theorem ordtcnv 16947
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 2296 . . . . . . . 8  |-  dom  R  =  dom  R
21psrn 14334 . . . . . . 7  |-  ( R  e.  PosetRel  ->  dom  R  =  ran  R )
32eqcomd 2301 . . . . . 6  |-  ( R  e.  PosetRel  ->  ran  R  =  dom  R )
43sneqd 3666 . . . . 5  |-  ( R  e.  PosetRel  ->  { ran  R }  =  { dom  R } )
5 vex 2804 . . . . . . . . . . . . 13  |-  y  e. 
_V
6 vex 2804 . . . . . . . . . . . . 13  |-  x  e. 
_V
75, 6brcnv 4880 . . . . . . . . . . . 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 2796 . . . . . . . . 9  |-  ( R  e.  PosetRel  ->  { y  e. 
ran  R  |  -.  y `' R x }  =  { y  e.  dom  R  |  -.  x R y } )
113, 10mpteq12dv 4114 . . . . . . . 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 4922 . . . . . . 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 4880 . . . . . . . . . . . 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 2796 . . . . . . . . 9  |-  ( R  e.  PosetRel  ->  { y  e. 
ran  R  |  -.  x `' R y }  =  { y  e.  dom  R  |  -.  y R x } )
173, 16mpteq12dv 4114 . . . . . . . 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 4922 . . . . . . 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 3343 . . . . . 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 3332 . . . . . 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 2344 . . . . 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 3343 . . . 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 5545 . . 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 5545 . 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 14337 . . 3  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
26 df-rn 4716 . . . 4  |-  ran  R  =  dom  `' R
27 eqid 2296 . . . 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 2296 . . . 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 16935 . . 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 2296 . . 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 2296 . . 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 16935 . 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 2338 1  |-  ( R  e.  PosetRel  ->  (ordTop `  `' R
)  =  (ordTop `  R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   {crab 2560    u. cun 3163   {csn 3653   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   dom cdm 4705   ran crn 4706   ` cfv 5271   ficfi 7180   topGenctg 13358  ordTopcordt 13414   PosetRelcps 14317
This theorem is referenced by:  ordtrest2  16950  cnvordtrestixx  23312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279  df-ordt 13418  df-ps 14322
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