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Theorem ordtcnvNEW 31584
Description: The order dual generates the same topology as the original order. (Contributed by Mario Carneiro, 3-Sep-2015.) (Revised by Thierry Arnoux, 13-Sep-2018.)
Hypotheses
Ref Expression
ordtNEW.b 𝐵 = (Base‘𝐾)
ordtNEW.l = ((le‘𝐾) ∩ (𝐵 × 𝐵))
Assertion
Ref Expression
ordtcnvNEW (𝐾 ∈ Proset → (ordTop‘ ) = (ordTop‘ ))

Proof of Theorem ordtcnvNEW
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3412 . . . . . . . . . . . . 13 𝑦 ∈ V
2 vex 3412 . . . . . . . . . . . . 13 𝑥 ∈ V
31, 2brcnv 5751 . . . . . . . . . . . 12 (𝑦 𝑥𝑥 𝑦)
43a1i 11 . . . . . . . . . . 11 (𝐾 ∈ Proset → (𝑦 𝑥𝑥 𝑦))
54notbid 321 . . . . . . . . . 10 (𝐾 ∈ Proset → (¬ 𝑦 𝑥 ↔ ¬ 𝑥 𝑦))
65rabbidv 3390 . . . . . . . . 9 (𝐾 ∈ Proset → {𝑦𝐵 ∣ ¬ 𝑦 𝑥} = {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
76mpteq2dv 5151 . . . . . . . 8 (𝐾 ∈ Proset → (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
87rneqd 5807 . . . . . . 7 (𝐾 ∈ Proset → ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
92, 1brcnv 5751 . . . . . . . . . . . 12 (𝑥 𝑦𝑦 𝑥)
109a1i 11 . . . . . . . . . . 11 (𝐾 ∈ Proset → (𝑥 𝑦𝑦 𝑥))
1110notbid 321 . . . . . . . . . 10 (𝐾 ∈ Proset → (¬ 𝑥 𝑦 ↔ ¬ 𝑦 𝑥))
1211rabbidv 3390 . . . . . . . . 9 (𝐾 ∈ Proset → {𝑦𝐵 ∣ ¬ 𝑥 𝑦} = {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
1312mpteq2dv 5151 . . . . . . . 8 (𝐾 ∈ Proset → (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
1413rneqd 5807 . . . . . . 7 (𝐾 ∈ Proset → ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
158, 14uneq12d 4078 . . . . . 6 (𝐾 ∈ Proset → (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})) = (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})))
16 uncom 4067 . . . . . 6 (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})) = (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
1715, 16eqtrdi 2794 . . . . 5 (𝐾 ∈ Proset → (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})) = (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})))
1817uneq2d 4077 . . . 4 (𝐾 ∈ Proset → ({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))) = ({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))
1918fveq2d 6721 . . 3 (𝐾 ∈ Proset → (fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})))) = (fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})))))
2019fveq2d 6721 . 2 (𝐾 ∈ Proset → (topGen‘(fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))))
21 eqid 2737 . . . 4 (ODual‘𝐾) = (ODual‘𝐾)
2221oduprs 30961 . . 3 (𝐾 ∈ Proset → (ODual‘𝐾) ∈ Proset )
23 ordtNEW.b . . . . 5 𝐵 = (Base‘𝐾)
2421, 23odubas 17799 . . . 4 𝐵 = (Base‘(ODual‘𝐾))
25 ordtNEW.l . . . . . 6 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
2625cnveqi 5743 . . . . 5 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
27 cnvin 6008 . . . . 5 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘𝐾) ∩ (𝐵 × 𝐵))
28 eqid 2737 . . . . . . 7 (le‘𝐾) = (le‘𝐾)
2921, 28oduleval 17797 . . . . . 6 (le‘𝐾) = (le‘(ODual‘𝐾))
30 cnvxp 6020 . . . . . 6 (𝐵 × 𝐵) = (𝐵 × 𝐵)
3129, 30ineq12i 4125 . . . . 5 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
3226, 27, 313eqtri 2769 . . . 4 = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
33 eqid 2737 . . . 4 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
34 eqid 2737 . . . 4 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
3524, 32, 33, 34ordtprsval 31582 . . 3 ((ODual‘𝐾) ∈ Proset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))))
3622, 35syl 17 . 2 (𝐾 ∈ Proset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))))
37 eqid 2737 . . 3 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
38 eqid 2737 . . 3 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
3923, 25, 37, 38ordtprsval 31582 . 2 (𝐾 ∈ Proset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))))
4020, 36, 393eqtr4d 2787 1 (𝐾 ∈ Proset → (ordTop‘ ) = (ordTop‘ ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209   = wceq 1543  wcel 2110  {crab 3065  cun 3864  cin 3865  {csn 4541   class class class wbr 5053  cmpt 5135   × cxp 5549  ccnv 5550  ran crn 5552  cfv 6380  ficfi 9026  Basecbs 16760  lecple 16809  topGenctg 16942  ordTopcordt 17004  ODualcodu 17794   Proset cproset 17800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-dec 12294  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ple 16822  df-ordt 17006  df-odu 17795  df-proset 17802
This theorem is referenced by:  ordtrest2NEW  31587
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