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Theorem ordtcnv 18764
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 2441 . . . . . . . 8  |-  dom  R  =  dom  R
21psrn 15375 . . . . . . 7  |-  ( R  e.  PosetRel  ->  dom  R  =  ran  R )
32eqcomd 2446 . . . . . 6  |-  ( R  e.  PosetRel  ->  ran  R  =  dom  R )
43sneqd 3886 . . . . 5  |-  ( R  e.  PosetRel  ->  { ran  R }  =  { dom  R } )
5 vex 2973 . . . . . . . . . . . . 13  |-  y  e. 
_V
6 vex 2973 . . . . . . . . . . . . 13  |-  x  e. 
_V
75, 6brcnv 5018 . . . . . . . . . . . 12  |-  ( y `' R x  <->  x R
y )
87a1i 11 . . . . . . . . . . 11  |-  ( R  e.  PosetRel  ->  ( y `' R x  <->  x R
y ) )
98notbid 294 . . . . . . . . . 10  |-  ( R  e.  PosetRel  ->  ( -.  y `' R x  <->  -.  x R y ) )
103, 9rabeqbidv 2965 . . . . . . . . 9  |-  ( R  e.  PosetRel  ->  { y  e. 
ran  R  |  -.  y `' R x }  =  { y  e.  dom  R  |  -.  x R y } )
113, 10mpteq12dv 4367 . . . . . . . 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 5063 . . . . . . 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 5018 . . . . . . . . . . . 12  |-  ( x `' R y  <->  y R x )
1413a1i 11 . . . . . . . . . . 11  |-  ( R  e.  PosetRel  ->  ( x `' R y  <->  y R x ) )
1514notbid 294 . . . . . . . . . 10  |-  ( R  e.  PosetRel  ->  ( -.  x `' R y  <->  -.  y R x ) )
163, 15rabeqbidv 2965 . . . . . . . . 9  |-  ( R  e.  PosetRel  ->  { y  e. 
ran  R  |  -.  x `' R y }  =  { y  e.  dom  R  |  -.  y R x } )
173, 16mpteq12dv 4367 . . . . . . . 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 5063 . . . . . . 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 3508 . . . . . 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 3497 . . . . . 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 2489 . . . . 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 3508 . . . 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 5692 . . 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 5692 . 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 15378 . . 3  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
26 df-rn 4847 . . . 4  |-  ran  R  =  dom  `' R
27 eqid 2441 . . . 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 2441 . . . 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 18752 . . 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 2441 . . 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 2441 . . 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 18752 . 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 2483 1  |-  ( R  e.  PosetRel  ->  (ordTop `  `' R
)  =  (ordTop `  R ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    = wceq 1364    e. wcel 1761   {crab 2717    u. cun 3323   {csn 3874   class class class wbr 4289    e. cmpt 4347   `'ccnv 4835   dom cdm 4836   ran crn 4837   ` cfv 5415   ficfi 7656   topGenctg 14372  ordTopcordt 14433   PosetRelcps 15364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-iota 5378  df-fun 5417  df-fv 5423  df-ordt 14435  df-ps 15366
This theorem is referenced by:  ordtrest2  18767  cnvordtrestixx  26279
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