MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordtcnv Structured version   Visualization version   GIF version

Theorem ordtcnv 22705
Description: The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
ordtcnv (𝑅 ∈ PosetRel β†’ (ordTopβ€˜β—‘π‘…) = (ordTopβ€˜π‘…))

Proof of Theorem ordtcnv
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . . . . . 8 dom 𝑅 = dom 𝑅
21psrn 18528 . . . . . . 7 (𝑅 ∈ PosetRel β†’ dom 𝑅 = ran 𝑅)
32eqcomd 2739 . . . . . 6 (𝑅 ∈ PosetRel β†’ ran 𝑅 = dom 𝑅)
43sneqd 4641 . . . . 5 (𝑅 ∈ PosetRel β†’ {ran 𝑅} = {dom 𝑅})
5 vex 3479 . . . . . . . . . . . . 13 𝑦 ∈ V
6 vex 3479 . . . . . . . . . . . . 13 π‘₯ ∈ V
75, 6brcnv 5883 . . . . . . . . . . . 12 (𝑦◑𝑅π‘₯ ↔ π‘₯𝑅𝑦)
87a1i 11 . . . . . . . . . . 11 (𝑅 ∈ PosetRel β†’ (𝑦◑𝑅π‘₯ ↔ π‘₯𝑅𝑦))
98notbid 318 . . . . . . . . . 10 (𝑅 ∈ PosetRel β†’ (Β¬ 𝑦◑𝑅π‘₯ ↔ Β¬ π‘₯𝑅𝑦))
103, 9rabeqbidv 3450 . . . . . . . . 9 (𝑅 ∈ PosetRel β†’ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯} = {𝑦 ∈ dom 𝑅 ∣ Β¬ π‘₯𝑅𝑦})
113, 10mpteq12dv 5240 . . . . . . . 8 (𝑅 ∈ PosetRel β†’ (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯}) = (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ π‘₯𝑅𝑦}))
1211rneqd 5938 . . . . . . 7 (𝑅 ∈ PosetRel β†’ ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯}) = ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ π‘₯𝑅𝑦}))
136, 5brcnv 5883 . . . . . . . . . . . 12 (π‘₯◑𝑅𝑦 ↔ 𝑦𝑅π‘₯)
1413a1i 11 . . . . . . . . . . 11 (𝑅 ∈ PosetRel β†’ (π‘₯◑𝑅𝑦 ↔ 𝑦𝑅π‘₯))
1514notbid 318 . . . . . . . . . 10 (𝑅 ∈ PosetRel β†’ (Β¬ π‘₯◑𝑅𝑦 ↔ Β¬ 𝑦𝑅π‘₯))
163, 15rabeqbidv 3450 . . . . . . . . 9 (𝑅 ∈ PosetRel β†’ {𝑦 ∈ ran 𝑅 ∣ Β¬ π‘₯◑𝑅𝑦} = {𝑦 ∈ dom 𝑅 ∣ Β¬ 𝑦𝑅π‘₯})
173, 16mpteq12dv 5240 . . . . . . . 8 (𝑅 ∈ PosetRel β†’ (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ π‘₯◑𝑅𝑦}) = (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ 𝑦𝑅π‘₯}))
1817rneqd 5938 . . . . . . 7 (𝑅 ∈ PosetRel β†’ ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ π‘₯◑𝑅𝑦}) = ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ 𝑦𝑅π‘₯}))
1912, 18uneq12d 4165 . . . . . 6 (𝑅 ∈ PosetRel β†’ (ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯}) βˆͺ ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ π‘₯◑𝑅𝑦})) = (ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ 𝑦𝑅π‘₯})))
20 uncom 4154 . . . . . 6 (ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ π‘₯𝑅𝑦}) βˆͺ ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ 𝑦𝑅π‘₯})) = (ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ 𝑦𝑅π‘₯}) βˆͺ ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ π‘₯𝑅𝑦}))
2119, 20eqtrdi 2789 . . . . 5 (𝑅 ∈ PosetRel β†’ (ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯}) βˆͺ ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ π‘₯◑𝑅𝑦})) = (ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ 𝑦𝑅π‘₯}) βˆͺ ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ π‘₯𝑅𝑦})))
224, 21uneq12d 4165 . . . 4 (𝑅 ∈ PosetRel β†’ ({ran 𝑅} βˆͺ (ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯}) βˆͺ ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ π‘₯◑𝑅𝑦}))) = ({dom 𝑅} βˆͺ (ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ 𝑦𝑅π‘₯}) βˆͺ ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ π‘₯𝑅𝑦}))))
2322fveq2d 6896 . . 3 (𝑅 ∈ PosetRel β†’ (fiβ€˜({ran 𝑅} βˆͺ (ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯}) βˆͺ ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ π‘₯◑𝑅𝑦})))) = (fiβ€˜({dom 𝑅} βˆͺ (ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ 𝑦𝑅π‘₯}) βˆͺ ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ π‘₯𝑅𝑦})))))
2423fveq2d 6896 . 2 (𝑅 ∈ PosetRel β†’ (topGenβ€˜(fiβ€˜({ran 𝑅} βˆͺ (ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯}) βˆͺ ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ π‘₯◑𝑅𝑦}))))) = (topGenβ€˜(fiβ€˜({dom 𝑅} βˆͺ (ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ 𝑦𝑅π‘₯}) βˆͺ ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ π‘₯𝑅𝑦}))))))
25 cnvps 18531 . . 3 (𝑅 ∈ PosetRel β†’ ◑𝑅 ∈ PosetRel)
26 df-rn 5688 . . . 4 ran 𝑅 = dom ◑𝑅
27 eqid 2733 . . . 4 ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯}) = ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯})
28 eqid 2733 . . . 4 ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ π‘₯◑𝑅𝑦}) = ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ π‘₯◑𝑅𝑦})
2926, 27, 28ordtval 22693 . . 3 (◑𝑅 ∈ PosetRel β†’ (ordTopβ€˜β—‘π‘…) = (topGenβ€˜(fiβ€˜({ran 𝑅} βˆͺ (ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯}) βˆͺ ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ π‘₯◑𝑅𝑦}))))))
3025, 29syl 17 . 2 (𝑅 ∈ PosetRel β†’ (ordTopβ€˜β—‘π‘…) = (topGenβ€˜(fiβ€˜({ran 𝑅} βˆͺ (ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ 𝑦◑𝑅π‘₯}) βˆͺ ran (π‘₯ ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ Β¬ π‘₯◑𝑅𝑦}))))))
31 eqid 2733 . . 3 ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ 𝑦𝑅π‘₯}) = ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ 𝑦𝑅π‘₯})
32 eqid 2733 . . 3 ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ π‘₯𝑅𝑦}) = ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ π‘₯𝑅𝑦})
331, 31, 32ordtval 22693 . 2 (𝑅 ∈ PosetRel β†’ (ordTopβ€˜π‘…) = (topGenβ€˜(fiβ€˜({dom 𝑅} βˆͺ (ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ 𝑦𝑅π‘₯}) βˆͺ ran (π‘₯ ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ Β¬ π‘₯𝑅𝑦}))))))
3424, 30, 333eqtr4d 2783 1 (𝑅 ∈ PosetRel β†’ (ordTopβ€˜β—‘π‘…) = (ordTopβ€˜π‘…))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107  {crab 3433   βˆͺ cun 3947  {csn 4629   class class class wbr 5149   ↦ cmpt 5232  β—‘ccnv 5676  dom cdm 5677  ran crn 5678  β€˜cfv 6544  ficfi 9405  topGenctg 17383  ordTopcordt 17445  PosetRelcps 18517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fv 6552  df-ordt 17447  df-ps 18519
This theorem is referenced by:  ordtrest2  22708  cnvordtrestixx  32893
  Copyright terms: Public domain W3C validator