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Theorem ordtopn2 20922
Description: A downward ray (-∞, 𝑃) is open. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ordttopon.3 𝑋 = dom 𝑅
Assertion
Ref Expression
ordtopn2 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ (ordTop‘𝑅))
Distinct variable groups:   𝑥,𝑃   𝑥,𝑅   𝑥,𝑉   𝑥,𝑋

Proof of Theorem ordtopn2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ordttopon.3 . . . . . . . . 9 𝑋 = dom 𝑅
2 eqid 2621 . . . . . . . . 9 ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) = ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦})
3 eqid 2621 . . . . . . . . 9 ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}) = ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})
41, 2, 3ordtuni 20917 . . . . . . . 8 (𝑅𝑉𝑋 = ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
54adantr 481 . . . . . . 7 ((𝑅𝑉𝑃𝑋) → 𝑋 = ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
6 dmexg 7051 . . . . . . . . 9 (𝑅𝑉 → dom 𝑅 ∈ V)
71, 6syl5eqel 2702 . . . . . . . 8 (𝑅𝑉𝑋 ∈ V)
87adantr 481 . . . . . . 7 ((𝑅𝑉𝑃𝑋) → 𝑋 ∈ V)
95, 8eqeltrrd 2699 . . . . . 6 ((𝑅𝑉𝑃𝑋) → ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
10 uniexb 6928 . . . . . 6 (({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V ↔ ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
119, 10sylibr 224 . . . . 5 ((𝑅𝑉𝑃𝑋) → ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V)
12 ssfii 8277 . . . . 5 (({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ∈ V → ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ⊆ (fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))))
1311, 12syl 17 . . . 4 ((𝑅𝑉𝑃𝑋) → ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ⊆ (fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))))
14 fibas 20705 . . . . 5 (fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))) ∈ TopBases
15 bastg 20694 . . . . 5 ((fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))) ∈ TopBases → (fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))) ⊆ (topGen‘(fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))))
1614, 15ax-mp 5 . . . 4 (fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))) ⊆ (topGen‘(fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))))
1713, 16syl6ss 3599 . . 3 ((𝑅𝑉𝑃𝑋) → ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ⊆ (topGen‘(fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))))
181, 2, 3ordtval 20916 . . . 4 (𝑅𝑉 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))))
1918adantr 481 . . 3 ((𝑅𝑉𝑃𝑋) → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))))
2017, 19sseqtr4d 3626 . 2 ((𝑅𝑉𝑃𝑋) → ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))) ⊆ (ordTop‘𝑅))
21 ssun2 3760 . . 3 (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})) ⊆ ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))
22 ssun2 3760 . . . 4 ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))
23 simpr 477 . . . . . 6 ((𝑅𝑉𝑃𝑋) → 𝑃𝑋)
24 eqidd 2622 . . . . . 6 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥})
25 breq1 4621 . . . . . . . . . 10 (𝑦 = 𝑃 → (𝑦𝑅𝑥𝑃𝑅𝑥))
2625notbid 308 . . . . . . . . 9 (𝑦 = 𝑃 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑃𝑅𝑥))
2726rabbidv 3180 . . . . . . . 8 (𝑦 = 𝑃 → {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥})
2827eqeq2d 2631 . . . . . . 7 (𝑦 = 𝑃 → ({𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥} ↔ {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥}))
2928rspcev 3298 . . . . . 6 ((𝑃𝑋 ∧ {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥}) → ∃𝑦𝑋 {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})
3023, 24, 29syl2anc 692 . . . . 5 ((𝑅𝑉𝑃𝑋) → ∃𝑦𝑋 {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})
31 rabexg 4777 . . . . . 6 (𝑋 ∈ V → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ V)
32 eqid 2621 . . . . . . 7 (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}) = (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})
3332elrnmpt 5337 . . . . . 6 ({𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ V → ({𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑦𝑋 {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))
348, 31, 333syl 18 . . . . 5 ((𝑅𝑉𝑃𝑋) → ({𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}) ↔ ∃𝑦𝑋 {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} = {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))
3530, 34mpbird 247 . . . 4 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))
3622, 35sseldi 3585 . . 3 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥})))
3721, 36sseldi 3585 . 2 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ ({𝑋} ∪ (ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑥𝑅𝑦}) ∪ ran (𝑦𝑋 ↦ {𝑥𝑋 ∣ ¬ 𝑦𝑅𝑥}))))
3820, 37sseldd 3588 1 ((𝑅𝑉𝑃𝑋) → {𝑥𝑋 ∣ ¬ 𝑃𝑅𝑥} ∈ (ordTop‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  {crab 2911  Vcvv 3189  cun 3557  wss 3559  {csn 4153   cuni 4407   class class class wbr 4618  cmpt 4678  dom cdm 5079  ran crn 5080  cfv 5852  ficfi 8268  topGenctg 16030  ordTopcordt 16091  TopBasesctb 20673
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 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7908  df-fin 7911  df-fi 8269  df-topgen 16036  df-ordt 16093  df-bases 20674
This theorem is referenced by:  ordtopn3  20923  ordtcld2  20925  ordtrest  20929  ordthauslem  21110  ordthmeolem  21527  ordtrestNEW  29773
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