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Theorem ordtprsuni 30093
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 30088 . . . . 5 (𝐾 ∈ Preset → dom = 𝐵)
43sneqd 4222 . . . 4 (𝐾 ∈ Preset → {dom } = {𝐵})
5 biidd 252 . . . . . . . 8 (𝐾 ∈ Preset → (¬ 𝑦 𝑥 ↔ ¬ 𝑦 𝑥))
63, 5rabeqbidv 3226 . . . . . . 7 (𝐾 ∈ Preset → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} = {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
73, 6mpteq12dv 4766 . . . . . 6 (𝐾 ∈ Preset → (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
87rneqd 5385 . . . . 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 3226 . . . . . . 7 (𝐾 ∈ Preset → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} = {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
113, 10mpteq12dv 4766 . . . . . 6 (𝐾 ∈ Preset → (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
1211rneqd 5385 . . . . 5 (𝐾 ∈ Preset → ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
138, 12uneq12d 3801 . . . 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 3801 . . 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 4478 . 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 6239 . . . . . 6 (le‘𝐾) ∈ V
1716inex1 4832 . . . . 5 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
182, 17eqeltri 2726 . . . 4 ∈ V
19 eqid 2651 . . . . 5 dom = dom
20 eqid 2651 . . . . 5 ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥})
21 eqid 2651 . . . . 5 ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦})
2219, 20, 21ordtuni 21042 . . . 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 2700 . 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 3798 . . . . 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 3800 . . 3 (𝐾 ∈ Preset → ({𝐵} ∪ (𝐸𝐹)) = ({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))
3029unieqd 4478 . 2 (𝐾 ∈ Preset → ({𝐵} ∪ (𝐸𝐹)) = ({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))
3115, 24, 303eqtr4d 2695 1 (𝐾 ∈ Preset → 𝐵 = ({𝐵} ∪ (𝐸𝐹)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  cun 3605  cin 3606  {csn 4210   cuni 4468   class class class wbr 4685  cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144  cfv 5926  Basecbs 15904  lecple 15995   Preset cpreset 16973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-preset 16975
This theorem is referenced by:  ordtrest2NEW  30097
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