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

Theorem ordtprsval 30299
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 6428 . . . . 5 (le‘𝐾) ∈ V
32inex1 5005 . . . 4 ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
41, 3eqeltri 2892 . . 3 ∈ V
5 eqid 2817 . . . 4 dom = dom
6 eqid 2817 . . . 4 ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥})
7 eqid 2817 . . . 4 ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦})
85, 6, 7ordtval 21215 . . 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 30295 . . . . . 6 (𝐾 ∈ Proset → dom = 𝐵)
1211sneqd 4393 . . . . 5 (𝐾 ∈ Proset → {dom } = {𝐵})
13 rabeq 3393 . . . . . . . . . 10 (dom = 𝐵 → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} = {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
1411, 13syl 17 . . . . . . . . 9 (𝐾 ∈ Proset → {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥} = {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
1511, 14mpteq12dv 4938 . . . . . . . 8 (𝐾 ∈ Proset → (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
1615rneqd 5565 . . . . . . 7 (𝐾 ∈ Proset → ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥}))
17 ordtposval.e . . . . . . 7 𝐸 = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑦 𝑥})
1816, 17syl6eqr 2869 . . . . . 6 (𝐾 ∈ Proset → ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) = 𝐸)
19 rabeq 3393 . . . . . . . . . 10 (dom = 𝐵 → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} = {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
2011, 19syl 17 . . . . . . . . 9 (𝐾 ∈ Proset → {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦} = {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
2111, 20mpteq12dv 4938 . . . . . . . 8 (𝐾 ∈ Proset → (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
2221rneqd 5565 . . . . . . 7 (𝐾 ∈ Proset → ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦}))
23 ordtposval.f . . . . . . 7 𝐹 = ran (𝑥𝐵 ↦ {𝑦𝐵 ∣ ¬ 𝑥 𝑦})
2422, 23syl6eqr 2869 . . . . . 6 (𝐾 ∈ Proset → ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}) = 𝐹)
2518, 24uneq12d 3978 . . . . 5 (𝐾 ∈ Proset → (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦})) = (𝐸𝐹))
2612, 25uneq12d 3978 . . . 4 (𝐾 ∈ Proset → ({dom } ∪ (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}))) = ({𝐵} ∪ (𝐸𝐹)))
2726fveq2d 6419 . . 3 (𝐾 ∈ Proset → (fi‘({dom } ∪ (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦})))) = (fi‘({𝐵} ∪ (𝐸𝐹))))
2827fveq2d 6419 . 2 (𝐾 ∈ Proset → (topGen‘(fi‘({dom } ∪ (ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑦 𝑥}) ∪ ran (𝑥 ∈ dom ↦ {𝑦 ∈ dom ∣ ¬ 𝑥 𝑦}))))) = (topGen‘(fi‘({𝐵} ∪ (𝐸𝐹)))))
299, 28syl5eq 2863 1 (𝐾 ∈ Proset → (ordTop‘ ) = (topGen‘(fi‘({𝐵} ∪ (𝐸𝐹)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1637  wcel 2157  {crab 3111  Vcvv 3402  cun 3778  cin 3779  {csn 4381   class class class wbr 4855  cmpt 4934   × cxp 5320  dom cdm 5322  ran crn 5323  cfv 6108  ficfi 8562  Basecbs 16075  lecple 16167  topGenctg 16310  ordTopcordt 16371   Proset cproset 17138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4986  ax-nul 4994  ax-pr 5107
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5230  df-xp 5328  df-rel 5329  df-cnv 5330  df-co 5331  df-dm 5332  df-rn 5333  df-iota 6071  df-fun 6110  df-fv 6116  df-ordt 16373  df-proset 17140
This theorem is referenced by:  ordtcnvNEW  30301  ordtrest2NEW  30304  ordtconnlem1  30305
  Copyright terms: Public domain W3C validator