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Theorem ordtprsval 30783
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 6556 . . . . 5 (le‘𝐾) ∈ V
32inex1 5117 . . . 4 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
41, 3eqeltri 2879 . . 3 ∈ V
5 eqid 2795 . . . 4 dom = dom
6 eqid 2795 . . . 4 ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥})
7 eqid 2795 . . . 4 ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦})
85, 6, 7ordtval 21486 . . 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 30779 . . . . . 6 (𝐾 ∈ Proset → dom = 𝐵)
1211sneqd 4488 . . . . 5 (𝐾 ∈ Proset → {dom } = {𝐵})
1311rabeqdv 3429 . . . . . . . . 9 (𝐾 ∈ Proset → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} = {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
1411, 13mpteq12dv 5050 . . . . . . . 8 (𝐾 ∈ Proset → (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
1514rneqd 5695 . . . . . . 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, 16syl6eqr 2849 . . . . . 6 (𝐾 ∈ Proset → ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = 𝐸)
1811rabeqdv 3429 . . . . . . . . 9 (𝐾 ∈ Proset → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} = {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
1911, 18mpteq12dv 5050 . . . . . . . 8 (𝐾 ∈ Proset → (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
2019rneqd 5695 . . . . . . 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, 21syl6eqr 2849 . . . . . 6 (𝐾 ∈ Proset → ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = 𝐹)
2317, 22uneq12d 4065 . . . . 5 (𝐾 ∈ Proset → (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦})) = (𝐸𝐹))
2412, 23uneq12d 4065 . . . 4 (𝐾 ∈ Proset → ({dom } ∪ (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}))) = ({𝐵} ∪ (𝐸𝐹)))
2524fveq2d 6547 . . 3 (𝐾 ∈ Proset → (fi‘({dom } ∪ (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦})))) = (fi‘({𝐵} ∪ (𝐸𝐹))))
2625fveq2d 6547 . 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, 26syl5eq 2843 1 (𝐾 ∈ Proset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (𝐸𝐹)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1522  wcel 2081  {crab 3109  Vcvv 3437  cun 3861  cin 3862  {csn 4476   class class class wbr 4966  cmpt 5045   × cxp 5446  dom cdm 5448  ran crn 5449  cfv 6230  ficfi 8725  Basecbs 16317  lecple 16406  topGenctg 16545  ordTopcordt 16606   Proset cproset 17370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pr 5226
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3710  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-br 4967  df-opab 5029  df-mpt 5046  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-iota 6194  df-fun 6232  df-fv 6238  df-ordt 16608  df-proset 17372
This theorem is referenced by:  ordtcnvNEW  30785  ordtrest2NEW  30788  ordtconnlem1  30789
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