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Theorem ordtprsval 31868
Description: Value of the order topology for a proset. (Contributed by Thierry Arnoux, 11-Sep-2015.)
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
ordtprsval (𝐾 ∈ Proset → (ordTop‘ ) = (topGen‘(fi‘({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 ordtprsval
StepHypRef Expression
1 ordtNEW.l . . . 4 = ((le‘𝐾) ∩ (𝐵 × 𝐵))
2 fvex 6787 . . . . 5 (le‘𝐾) ∈ V
32inex1 5241 . . . 4 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
41, 3eqeltri 2835 . . 3 ∈ V
5 eqid 2738 . . . 4 dom = dom
6 eqid 2738 . . . 4 ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥})
7 eqid 2738 . . . 4 ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦})
85, 6, 7ordtval 22340 . . 3 ( ∈ V → (ordTop‘ ) = (topGen‘(fi‘({dom } ∪ (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}))))))
94, 8ax-mp 5 . 2 (ordTop‘ ) = (topGen‘(fi‘({dom } ∪ (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦})))))
10 ordtNEW.b . . . . . . 7 𝐵 = (Base‘𝐾)
1110, 1prsdm 31864 . . . . . 6 (𝐾 ∈ Proset → dom = 𝐵)
1211sneqd 4573 . . . . 5 (𝐾 ∈ Proset → {dom } = {𝐵})
1311rabeqdv 3419 . . . . . . . . 9 (𝐾 ∈ Proset → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} = {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
1411, 13mpteq12dv 5165 . . . . . . . 8 (𝐾 ∈ Proset → (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
1514rneqd 5847 . . . . . . 7 (𝐾 ∈ Proset → ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
16 ordtposval.e . . . . . . 7 𝐸 = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
1715, 16eqtr4di 2796 . . . . . 6 (𝐾 ∈ Proset → ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = 𝐸)
1811rabeqdv 3419 . . . . . . . . 9 (𝐾 ∈ Proset → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} = {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
1911, 18mpteq12dv 5165 . . . . . . . 8 (𝐾 ∈ Proset → (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
2019rneqd 5847 . . . . . . 7 (𝐾 ∈ Proset → ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
21 ordtposval.f . . . . . . 7 𝐹 = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
2220, 21eqtr4di 2796 . . . . . 6 (𝐾 ∈ Proset → ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = 𝐹)
2317, 22uneq12d 4098 . . . . 5 (𝐾 ∈ Proset → (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦})) = (𝐸𝐹))
2412, 23uneq12d 4098 . . . 4 (𝐾 ∈ Proset → ({dom } ∪ (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}))) = ({𝐵} ∪ (𝐸𝐹)))
2524fveq2d 6778 . . 3 (𝐾 ∈ Proset → (fi‘({dom } ∪ (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦})))) = (fi‘({𝐵} ∪ (𝐸𝐹))))
2625fveq2d 6778 . 2 (𝐾 ∈ Proset → (topGen‘(fi‘({dom } ∪ (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}))))) = (topGen‘(fi‘({𝐵} ∪ (𝐸𝐹)))))
279, 26eqtrid 2790 1 (𝐾 ∈ Proset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (𝐸𝐹)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1539  wcel 2106  {crab 3068  Vcvv 3432  cun 3885  cin 3886  {csn 4561   class class class wbr 5074  cmpt 5157   × cxp 5587  dom cdm 5589  ran crn 5590  cfv 6433  ficfi 9169  Basecbs 16912  lecple 16969  topGenctg 17148  ordTopcordt 17210   Proset cproset 18011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fv 6441  df-ordt 17212  df-proset 18013
This theorem is referenced by:  ordtcnvNEW  31870  ordtrest2NEW  31873  ordtconnlem1  31874
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