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Theorem ordttopon 23141
Description: Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ordttopon.3 𝑋 = dom 𝑅
Assertion
Ref Expression
ordttopon (𝑅𝑉 → (ordTop‘𝑅) ∈ (TopOn‘𝑋))

Proof of Theorem ordttopon
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordttopon.3 . . . 4 𝑋 = dom 𝑅
2 eqid 2725 . . . 4 ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
3 eqid 2725 . . . 4 ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
41, 2, 3ordtval 23137 . . 3 (𝑅𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))))))
5 fibas 22924 . . . 4 (fi‘({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})))) ∈ TopBases
6 tgtopon 22918 . . . 4 ((fi‘({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})))) ∈ TopBases → (topGen‘(fi‘({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))))) ∈ (TopOn‘ (fi‘({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))))))
75, 6ax-mp 5 . . 3 (topGen‘(fi‘({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))))) ∈ (TopOn‘ (fi‘({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})))))
84, 7eqeltrdi 2833 . 2 (𝑅𝑉 → (ordTop‘𝑅) ∈ (TopOn‘ (fi‘({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))))))
91, 2, 3ordtuni 23138 . . . 4 (𝑅𝑉𝑋 = ({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))))
10 dmexg 7909 . . . . . . . 8 (𝑅𝑉 → dom 𝑅 ∈ V)
111, 10eqeltrid 2829 . . . . . . 7 (𝑅𝑉𝑋 ∈ V)
129, 11eqeltrrd 2826 . . . . . 6 (𝑅𝑉 ({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))) ∈ V)
13 uniexb 7767 . . . . . 6 (({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))) ∈ V ↔ ({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))) ∈ V)
1412, 13sylibr 233 . . . . 5 (𝑅𝑉 → ({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))) ∈ V)
15 fiuni 9453 . . . . 5 (({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))) ∈ V → ({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))) = (fi‘({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})))))
1614, 15syl 17 . . . 4 (𝑅𝑉 ({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))) = (fi‘({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})))))
179, 16eqtrd 2765 . . 3 (𝑅𝑉𝑋 = (fi‘({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})))))
1817fveq2d 6900 . 2 (𝑅𝑉 → (TopOn‘𝑋) = (TopOn‘ (fi‘({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))))))
198, 18eleqtrrd 2828 1 (𝑅𝑉 → (ordTop‘𝑅) ∈ (TopOn‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1533  wcel 2098  {crab 3418  Vcvv 3461  cun 3942  {csn 4630   cuni 4909   class class class wbr 5149  cmpt 5232  dom cdm 5678  ran crn 5679  cfv 6549  ficfi 9435  topGenctg 17422  ordTopcordt 17484  TopOnctopon 22856  TopBasesctb 22892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-om 7872  df-1o 8487  df-er 8725  df-en 8965  df-fin 8968  df-fi 9436  df-topgen 17428  df-ordt 17486  df-top 22840  df-topon 22857  df-bases 22893
This theorem is referenced by:  ordtopn3  23144  ordtcld1  23145  ordtcld2  23146  ordttop  23148  ordtrest  23150  ordtrest2lem  23151  ordtrest2  23152  letopon  23153  ordtt1  23327  ordthaus  23332  ordthmeolem  23749  ordtrestNEW  33650  ordtrest2NEWlem  33651  ordtrest2NEW  33652  ordtconnlem1  33653
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