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Theorem ordttopon 20910
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 2621 . . . 4 ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
3 eqid 2621 . . . 4 ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
41, 2, 3ordtval 20906 . . 3 (𝑅𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))))))
5 fibas 20695 . . . 4 (fi‘({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})))) ∈ TopBases
6 tgtopon 20689 . . . 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, 7syl6eqel 2706 . 2 (𝑅𝑉 → (ordTop‘𝑅) ∈ (TopOn‘ (fi‘({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))))))
91, 2, 3ordtuni 20907 . . . 4 (𝑅𝑉𝑋 = ({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))))
10 dmexg 7047 . . . . . . . 8 (𝑅𝑉 → dom 𝑅 ∈ V)
111, 10syl5eqel 2702 . . . . . . 7 (𝑅𝑉𝑋 ∈ V)
129, 11eqeltrrd 2699 . . . . . 6 (𝑅𝑉 ({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))) ∈ V)
13 uniexb 6924 . . . . . 6 (({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))) ∈ V ↔ ({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))) ∈ V)
1412, 13sylibr 224 . . . . 5 (𝑅𝑉 → ({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))) ∈ V)
15 fiuni 8281 . . . . 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 2655 . . 3 (𝑅𝑉𝑋 = (fi‘({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})))))
1817fveq2d 6154 . 2 (𝑅𝑉 → (TopOn‘𝑋) = (TopOn‘ (fi‘({𝑋} ∪ (ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥}) ∪ ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦}))))))
198, 18eleqtrrd 2701 1 (𝑅𝑉 → (ordTop‘𝑅) ∈ (TopOn‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  cun 3554  {csn 4150   cuni 4404   class class class wbr 4615  cmpt 4675  dom cdm 5076  ran crn 5077  cfv 5849  ficfi 8263  topGenctg 16022  ordTopcordt 16083  TopOnctopon 20637  TopBasesctb 20663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-oadd 7512  df-er 7690  df-en 7903  df-fin 7906  df-fi 8264  df-topgen 16028  df-ordt 16085  df-top 20621  df-topon 20638  df-bases 20664
This theorem is referenced by:  ordtopn3  20913  ordtcld1  20914  ordtcld2  20915  ordttop  20917  ordtrest  20919  ordtrest2lem  20920  ordtrest2  20921  letopon  20922  ordtt1  21096  ordthaus  21101  ordthmeolem  21517  ordtrestNEW  29761  ordtrest2NEWlem  29762  ordtrest2NEW  29763  ordtconnlem1  29764
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