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Theorem ordtcnv 21376
 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 → (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 2825 . . . . . . . 8 dom 𝑅 = dom 𝑅
21psrn 17562 . . . . . . 7 (𝑅 ∈ PosetRel → dom 𝑅 = ran 𝑅)
32eqcomd 2831 . . . . . 6 (𝑅 ∈ PosetRel → ran 𝑅 = dom 𝑅)
43sneqd 4409 . . . . 5 (𝑅 ∈ PosetRel → {ran 𝑅} = {dom 𝑅})
5 vex 3417 . . . . . . . . . . . . 13 𝑦 ∈ V
6 vex 3417 . . . . . . . . . . . . 13 𝑥 ∈ V
75, 6brcnv 5537 . . . . . . . . . . . 12 (𝑦𝑅𝑥𝑥𝑅𝑦)
87a1i 11 . . . . . . . . . . 11 (𝑅 ∈ PosetRel → (𝑦𝑅𝑥𝑥𝑅𝑦))
98notbid 310 . . . . . . . . . 10 (𝑅 ∈ PosetRel → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦))
103, 9rabeqbidv 3408 . . . . . . . . 9 (𝑅 ∈ PosetRel → {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥} = {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦})
113, 10mpteq12dv 4956 . . . . . . . 8 (𝑅 ∈ PosetRel → (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}) = (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}))
1211rneqd 5585 . . . . . . 7 (𝑅 ∈ PosetRel → ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}))
136, 5brcnv 5537 . . . . . . . . . . . 12 (𝑥𝑅𝑦𝑦𝑅𝑥)
1413a1i 11 . . . . . . . . . . 11 (𝑅 ∈ PosetRel → (𝑥𝑅𝑦𝑦𝑅𝑥))
1514notbid 310 . . . . . . . . . 10 (𝑅 ∈ PosetRel → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥))
163, 15rabeqbidv 3408 . . . . . . . . 9 (𝑅 ∈ PosetRel → {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥𝑅𝑦} = {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥})
173, 16mpteq12dv 4956 . . . . . . . 8 (𝑅 ∈ PosetRel → (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥𝑅𝑦}) = (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
1817rneqd 5585 . . . . . . 7 (𝑅 ∈ PosetRel → ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}))
1912, 18uneq12d 3995 . . . . . 6 (𝑅 ∈ PosetRel → (ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥𝑅𝑦})) = (ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥})))
20 uncom 3984 . . . . . 6 (ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥})) = (ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}))
2119, 20syl6eq 2877 . . . . 5 (𝑅 ∈ PosetRel → (ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥𝑅𝑦})) = (ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦})))
224, 21uneq12d 3995 . . . 4 (𝑅 ∈ PosetRel → ({ran 𝑅} ∪ (ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥𝑅𝑦}))) = ({dom 𝑅} ∪ (ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}))))
2322fveq2d 6437 . . 3 (𝑅 ∈ PosetRel → (fi‘({ran 𝑅} ∪ (ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥𝑅𝑦})))) = (fi‘({dom 𝑅} ∪ (ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦})))))
2423fveq2d 6437 . 2 (𝑅 ∈ PosetRel → (topGen‘(fi‘({ran 𝑅} ∪ (ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥𝑅𝑦}))))) = (topGen‘(fi‘({dom 𝑅} ∪ (ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}))))))
25 cnvps 17565 . . 3 (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)
26 df-rn 5353 . . . 4 ran 𝑅 = dom 𝑅
27 eqid 2825 . . . 4 ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥})
28 eqid 2825 . . . 4 ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥𝑅𝑦})
2926, 27, 28ordtval 21364 . . 3 (𝑅 ∈ PosetRel → (ordTop‘𝑅) = (topGen‘(fi‘({ran 𝑅} ∪ (ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥𝑅𝑦}))))))
3025, 29syl 17 . 2 (𝑅 ∈ PosetRel → (ordTop‘𝑅) = (topGen‘(fi‘({ran 𝑅} ∪ (ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ ran 𝑅 ↦ {𝑦 ∈ ran 𝑅 ∣ ¬ 𝑥𝑅𝑦}))))))
31 eqid 2825 . . 3 ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥})
32 eqid 2825 . . 3 ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦})
331, 31, 32ordtval 21364 . 2 (𝑅 ∈ PosetRel → (ordTop‘𝑅) = (topGen‘(fi‘({dom 𝑅} ∪ (ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥 ∈ dom 𝑅 ↦ {𝑦 ∈ dom 𝑅 ∣ ¬ 𝑥𝑅𝑦}))))))
3424, 30, 333eqtr4d 2871 1 (𝑅 ∈ PosetRel → (ordTop‘𝑅) = (ordTop‘𝑅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   = wceq 1658   ∈ wcel 2166  {crab 3121   ∪ cun 3796  {csn 4397   class class class wbr 4873   ↦ cmpt 4952  ◡ccnv 5341  dom cdm 5342  ran crn 5343  ‘cfv 6123  ficfi 8585  topGenctg 16451  ordTopcordt 16512  PosetRelcps 17551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-iota 6086  df-fun 6125  df-fv 6131  df-ordt 16514  df-ps 17553 This theorem is referenced by:  ordtrest2  21379  cnvordtrestixx  30504
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