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Theorem ordtcnvNEW 33735
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 3466 . . . . . . . . . . . . 13 𝑦 ∈ V
2 vex 3466 . . . . . . . . . . . . 13 𝑥 ∈ V
31, 2brcnv 5889 . . . . . . . . . . . 12 (𝑦 𝑥𝑥 𝑦)
43a1i 11 . . . . . . . . . . 11 (𝐾 ∈ Proset → (𝑦 𝑥𝑥 𝑦))
54notbid 317 . . . . . . . . . 10 (𝐾 ∈ Proset → (¬ 𝑦 𝑥 ↔ ¬ 𝑥 𝑦))
65rabbidv 3427 . . . . . . . . 9 (𝐾 ∈ Proset → {𝑦𝐵 ∣ ¬ 𝑦 𝑥} = {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
76mpteq2dv 5255 . . . . . . . 8 (𝐾 ∈ Proset → (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
87rneqd 5944 . . . . . . 7 (𝐾 ∈ Proset → ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
92, 1brcnv 5889 . . . . . . . . . . . 12 (𝑥 𝑦𝑦 𝑥)
109a1i 11 . . . . . . . . . . 11 (𝐾 ∈ Proset → (𝑥 𝑦𝑦 𝑥))
1110notbid 317 . . . . . . . . . 10 (𝐾 ∈ Proset → (¬ 𝑥 𝑦 ↔ ¬ 𝑦 𝑥))
1211rabbidv 3427 . . . . . . . . 9 (𝐾 ∈ Proset → {𝑦𝐵 ∣ ¬ 𝑥 𝑦} = {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
1312mpteq2dv 5255 . . . . . . . 8 (𝐾 ∈ Proset → (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
1413rneqd 5944 . . . . . . 7 (𝐾 ∈ Proset → ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
158, 14uneq12d 4164 . . . . . 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 4153 . . . . . 6 (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})) = (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
1715, 16eqtrdi 2782 . . . . 5 (𝐾 ∈ Proset → (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})) = (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})))
1817uneq2d 4163 . . . 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 6905 . . 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 6905 . 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 2726 . . . 4 (ODual‘𝐾) = (ODual‘𝐾)
2221oduprs 32834 . . 3 (𝐾 ∈ Proset → (ODual‘𝐾) ∈ Proset )
23 ordtNEW.b . . . . 5 𝐵 = (Base‘𝐾)
2421, 23odubas 18316 . . . 4 𝐵 = (Base‘(ODual‘𝐾))
25 ordtNEW.l . . . . . 6 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
2625cnveqi 5881 . . . . 5 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
27 cnvin 6156 . . . . 5 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘𝐾) ∩ (𝐵 × 𝐵))
28 eqid 2726 . . . . . . 7 (le‘𝐾) = (le‘𝐾)
2921, 28oduleval 18314 . . . . . 6 (le‘𝐾) = (le‘(ODual‘𝐾))
30 cnvxp 6168 . . . . . 6 (𝐵 × 𝐵) = (𝐵 × 𝐵)
3129, 30ineq12i 4211 . . . . 5 ((le‘𝐾) ∩ (𝐵 × 𝐵)) = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
3226, 27, 313eqtri 2758 . . . 4 = ((le‘(ODual‘𝐾)) ∩ (𝐵 × 𝐵))
33 eqid 2726 . . . 4 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
34 eqid 2726 . . . 4 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
3524, 32, 33, 34ordtprsval 33733 . . 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 2726 . . 3 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
38 eqid 2726 . . 3 ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
3923, 25, 37, 38ordtprsval 33733 . 2 (𝐾 ∈ Proset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))))
4020, 36, 393eqtr4d 2776 1 (𝐾 ∈ Proset → (ordTop‘ ) = (ordTop‘ ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1534  wcel 2099  {crab 3419  cun 3945  cin 3946  {csn 4633   class class class wbr 5153  cmpt 5236   × cxp 5680  ccnv 5681  ran crn 5683  cfv 6554  ficfi 9453  Basecbs 17213  lecple 17273  topGenctg 17452  ordTopcordt 17514  ODualcodu 18311   Proset cproset 18318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-dec 12730  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-ple 17286  df-ordt 17516  df-odu 18312  df-proset 18320
This theorem is referenced by:  ordtrest2NEW  33738
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