Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ordtprsuni Structured version   Visualization version   GIF version

Theorem ordtprsuni 31394
 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 ∈ Proset → 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 31389 . . . . 5 (𝐾 ∈ Proset → dom = 𝐵)
43sneqd 4537 . . . 4 (𝐾 ∈ Proset → {dom } = {𝐵})
5 biidd 265 . . . . . . . 8 (𝐾 ∈ Proset → (¬ 𝑦 𝑥 ↔ ¬ 𝑦 𝑥))
63, 5rabeqbidv 3398 . . . . . . 7 (𝐾 ∈ Proset → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} = {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
73, 6mpteq12dv 5120 . . . . . 6 (𝐾 ∈ Proset → (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
87rneqd 5783 . . . . 5 (𝐾 ∈ Proset → ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
9 biidd 265 . . . . . . . 8 (𝐾 ∈ Proset → (¬ 𝑥 𝑦 ↔ ¬ 𝑥 𝑦))
103, 9rabeqbidv 3398 . . . . . . 7 (𝐾 ∈ Proset → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} = {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
113, 10mpteq12dv 5120 . . . . . 6 (𝐾 ∈ Proset → (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
1211rneqd 5783 . . . . 5 (𝐾 ∈ Proset → ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
138, 12uneq12d 4071 . . . 4 (𝐾 ∈ Proset → (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 4071 . . 3 (𝐾 ∈ Proset → ({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 4815 . 2 (𝐾 ∈ Proset → ({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 6675 . . . . . 6 (le‘𝐾) ∈ V
1716inex1 5190 . . . . 5 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
182, 17eqeltri 2848 . . . 4 ∈ V
19 eqid 2758 . . . . 5 dom = dom
20 eqid 2758 . . . . 5 ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥})
21 eqid 2758 . . . . 5 ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦})
2219, 20, 21ordtuni 21895 . . . 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 2808 . 2 (𝐾 ∈ Proset → 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 4068 . . . . 5 (𝐸𝐹) = (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
2827a1i 11 . . . 4 (𝐾 ∈ Proset → (𝐸𝐹) = (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})))
2928uneq2d 4070 . . 3 (𝐾 ∈ Proset → ({𝐵} ∪ (𝐸𝐹)) = ({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))
3029unieqd 4815 . 2 (𝐾 ∈ Proset → ({𝐵} ∪ (𝐸𝐹)) = ({𝐵} ∪ (ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))))
3115, 24, 303eqtr4d 2803 1 (𝐾 ∈ Proset → 𝐵 = ({𝐵} ∪ (𝐸𝐹)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2111  {crab 3074  Vcvv 3409   ∪ cun 3858   ∩ cin 3859  {csn 4525  ∪ cuni 4801   class class class wbr 5035   ↦ cmpt 5115   × cxp 5525  dom cdm 5527  ran crn 5528  ‘cfv 6339  Basecbs 16546  lecple 16635   Proset cproset 17607 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-proset 17609 This theorem is referenced by:  ordtrest2NEW  31398
 Copyright terms: Public domain W3C validator