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Theorem ordtprsuni 29939
Description: Value of the order topology. (Contributed by Thierry Arnoux, 13-Sep-2018.)
Hypotheses
Ref Expression
ordtNEW.b 𝐵 = (Base‘𝐾)
ordtNEW.l = ((le‘𝐾) ∩ (𝐵 × 𝐵))
ordtposval.e 𝐸 = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
ordtposval.f 𝐹 = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
Assertion
Ref Expression
ordtprsuni (𝐾 ∈ Preset → 𝐵 = ({𝐵} ∪ (𝐸𝐹)))
Distinct variable groups:   𝑥,𝑦,   𝑥,𝐵,𝑦   𝑥,𝐾,𝑦
Allowed substitution hints:   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem ordtprsuni
StepHypRef Expression
1 ordtNEW.b . . . . . 6 𝐵 = (Base‘𝐾)
2 ordtNEW.l . . . . . 6 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
31, 2prsdm 29934 . . . . 5 (𝐾 ∈ Preset → dom = 𝐵)
43sneqd 4180 . . . 4 (𝐾 ∈ Preset → {dom } = {𝐵})
5 biidd 252 . . . . . . . 8 (𝐾 ∈ Preset → (¬ 𝑦 𝑥 ↔ ¬ 𝑦 𝑥))
63, 5rabeqbidv 3190 . . . . . . 7 (𝐾 ∈ Preset → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} = {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
73, 6mpteq12dv 4724 . . . . . 6 (𝐾 ∈ Preset → (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
87rneqd 5342 . . . . 5 (𝐾 ∈ Preset → ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
9 biidd 252 . . . . . . . 8 (𝐾 ∈ Preset → (¬ 𝑥 𝑦 ↔ ¬ 𝑥 𝑦))
103, 9rabeqbidv 3190 . . . . . . 7 (𝐾 ∈ Preset → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} = {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
113, 10mpteq12dv 4724 . . . . . 6 (𝐾 ∈ Preset → (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
1211rneqd 5342 . . . . 5 (𝐾 ∈ Preset → ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
138, 12uneq12d 3760 . . . 4 (𝐾 ∈ Preset → (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦})) = (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})))
144, 13uneq12d 3760 . . 3 (𝐾 ∈ Preset → ({dom } ∪ (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}))) = ({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))
1514unieqd 4437 . 2 (𝐾 ∈ Preset → ({dom } ∪ (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}))) = ({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))
16 fvex 6188 . . . . . 6 (le‘𝐾) ∈ V
1716inex1 4790 . . . . 5 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
182, 17eqeltri 2695 . . . 4 ∈ V
19 eqid 2620 . . . . 5 dom = dom
20 eqid 2620 . . . . 5 ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥})
21 eqid 2620 . . . . 5 ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦})
2219, 20, 21ordtuni 20975 . . . 4 ( ∈ V → dom = ({dom } ∪ (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}))))
2318, 22ax-mp 5 . . 3 dom = ({dom } ∪ (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦})))
2423, 3syl5reqr 2669 . 2 (𝐾 ∈ Preset → 𝐵 = ({dom } ∪ (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}))))
25 ordtposval.e . . . . . 6 𝐸 = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
26 ordtposval.f . . . . . 6 𝐹 = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
2725, 26uneq12i 3757 . . . . 5 (𝐸𝐹) = (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
2827a1i 11 . . . 4 (𝐾 ∈ Preset → (𝐸𝐹) = (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})))
2928uneq2d 3759 . . 3 (𝐾 ∈ Preset → ({𝐵} ∪ (𝐸𝐹)) = ({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))
3029unieqd 4437 . 2 (𝐾 ∈ Preset → ({𝐵} ∪ (𝐸𝐹)) = ({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))
3115, 24, 303eqtr4d 2664 1 (𝐾 ∈ Preset → 𝐵 = ({𝐵} ∪ (𝐸𝐹)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1481  wcel 1988  {crab 2913  Vcvv 3195  cun 3565  cin 3566  {csn 4168   cuni 4427   class class class wbr 4644  cmpt 4720   × cxp 5102  dom cdm 5104  ran crn 5105  cfv 5876  Basecbs 15838  lecple 15929   Preset cpreset 16907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-preset 16909
This theorem is referenced by:  ordtrest2NEW  29943
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