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Theorem ordtbas 20901
Description: In a total order, the finite intersections of the open rays generates the set of open intervals, but no more - these four collections form a subbasis for the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypotheses
Ref Expression
ordtval.1 𝑋 = dom 𝑅
ordtval.2 𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
ordtval.3 𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
ordtval.4 𝐶 = ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
Assertion
Ref Expression
ordtbas (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴𝐵))) = (({𝑋} ∪ (𝐴𝐵)) ∪ 𝐶))
Distinct variable groups:   𝑎,𝑏,𝐴   𝑥,𝑎,𝑦,𝑅,𝑏   𝑋,𝑎,𝑏,𝑥,𝑦   𝐵,𝑎,𝑏
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem ordtbas
Dummy variables 𝑚 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 4874 . . . . . 6 {𝑋} ∈ V
2 ssun2 3760 . . . . . . 7 (𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵))
3 ordtval.1 . . . . . . . . . 10 𝑋 = dom 𝑅
4 ordtval.2 . . . . . . . . . 10 𝐴 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑦𝑅𝑥})
5 ordtval.3 . . . . . . . . . 10 𝐵 = ran (𝑥𝑋 ↦ {𝑦𝑋 ∣ ¬ 𝑥𝑅𝑦})
63, 4, 5ordtuni 20899 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 = ({𝑋} ∪ (𝐴𝐵)))
7 dmexg 7045 . . . . . . . . . 10 (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
83, 7syl5eqel 2708 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 ∈ V)
96, 8eqeltrrd 2705 . . . . . . . 8 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
10 uniexb 6922 . . . . . . . 8 (({𝑋} ∪ (𝐴𝐵)) ∈ V ↔ ({𝑋} ∪ (𝐴𝐵)) ∈ V)
119, 10sylibr 224 . . . . . . 7 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
12 ssexg 4769 . . . . . . 7 (((𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵)) ∧ ({𝑋} ∪ (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
132, 11, 12sylancr 694 . . . . . 6 (𝑅 ∈ TosetRel → (𝐴𝐵) ∈ V)
14 elfiun 8281 . . . . . 6 (({𝑋} ∈ V ∧ (𝐴𝐵) ∈ V) → (𝑧 ∈ (fi‘({𝑋} ∪ (𝐴𝐵))) ↔ (𝑧 ∈ (fi‘{𝑋}) ∨ 𝑧 ∈ (fi‘(𝐴𝐵)) ∨ ∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴𝐵))𝑧 = (𝑚𝑛))))
151, 13, 14sylancr 694 . . . . 5 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘({𝑋} ∪ (𝐴𝐵))) ↔ (𝑧 ∈ (fi‘{𝑋}) ∨ 𝑧 ∈ (fi‘(𝐴𝐵)) ∨ ∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴𝐵))𝑧 = (𝑚𝑛))))
16 fisn 8278 . . . . . . . . 9 (fi‘{𝑋}) = {𝑋}
17 ssun1 3759 . . . . . . . . 9 {𝑋} ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
1816, 17eqsstri 3619 . . . . . . . 8 (fi‘{𝑋}) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
1918sseli 3584 . . . . . . 7 (𝑧 ∈ (fi‘{𝑋}) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
2019a1i 11 . . . . . 6 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘{𝑋}) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
21 ordtval.4 . . . . . . . . 9 𝐶 = ran (𝑎𝑋, 𝑏𝑋 ↦ {𝑦𝑋 ∣ (¬ 𝑦𝑅𝑎 ∧ ¬ 𝑏𝑅𝑦)})
223, 4, 5, 21ordtbas2 20900 . . . . . . . 8 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) = ((𝐴𝐵) ∪ 𝐶))
23 ssun2 3760 . . . . . . . 8 ((𝐴𝐵) ∪ 𝐶) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
2422, 23syl6eqss 3639 . . . . . . 7 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
2524sseld 3587 . . . . . 6 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴𝐵)) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
26 fipwuni 8277 . . . . . . . . . . . . . . 15 (fi‘(𝐴𝐵)) ⊆ 𝒫 (𝐴𝐵)
2726sseli 3584 . . . . . . . . . . . . . 14 (𝑛 ∈ (fi‘(𝐴𝐵)) → 𝑛 ∈ 𝒫 (𝐴𝐵))
2827elpwid 4146 . . . . . . . . . . . . 13 (𝑛 ∈ (fi‘(𝐴𝐵)) → 𝑛 (𝐴𝐵))
2928ad2antll 764 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛 (𝐴𝐵))
302unissi 4432 . . . . . . . . . . . . . 14 (𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵))
3130, 6syl5sseqr 3638 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → (𝐴𝐵) ⊆ 𝑋)
3231adantr 481 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝐴𝐵) ⊆ 𝑋)
3329, 32sstrd 3598 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛𝑋)
34 simprl 793 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑚 ∈ (fi‘{𝑋}))
3534, 16syl6eleq 2714 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑚 ∈ {𝑋})
36 elsni 4170 . . . . . . . . . . . 12 (𝑚 ∈ {𝑋} → 𝑚 = 𝑋)
3735, 36syl 17 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑚 = 𝑋)
3833, 37sseqtr4d 3626 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛𝑚)
39 sseqin2 3800 . . . . . . . . . 10 (𝑛𝑚 ↔ (𝑚𝑛) = 𝑛)
4038, 39sylib 208 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝑚𝑛) = 𝑛)
4124sselda 3588 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ 𝑛 ∈ (fi‘(𝐴𝐵))) → 𝑛 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
4241adantrl 751 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
4340, 42eqeltrd 2704 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝑚𝑛) ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
44 eleq1 2692 . . . . . . . 8 (𝑧 = (𝑚𝑛) → (𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)) ↔ (𝑚𝑛) ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4543, 44syl5ibrcom 237 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝑧 = (𝑚𝑛) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4645rexlimdvva 3036 . . . . . 6 (𝑅 ∈ TosetRel → (∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴𝐵))𝑧 = (𝑚𝑛) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4720, 25, 463jaod 1389 . . . . 5 (𝑅 ∈ TosetRel → ((𝑧 ∈ (fi‘{𝑋}) ∨ 𝑧 ∈ (fi‘(𝐴𝐵)) ∨ ∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴𝐵))𝑧 = (𝑚𝑛)) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4815, 47sylbid 230 . . . 4 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘({𝑋} ∪ (𝐴𝐵))) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4948ssrdv 3594 . . 3 (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴𝐵))) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
50 ssfii 8270 . . . . . 6 (({𝑋} ∪ (𝐴𝐵)) ∈ V → ({𝑋} ∪ (𝐴𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5111, 50syl 17 . . . . 5 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5251unssad 3773 . . . 4 (𝑅 ∈ TosetRel → {𝑋} ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
53 fiss 8275 . . . . . 6 ((({𝑋} ∪ (𝐴𝐵)) ∈ V ∧ (𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵))) → (fi‘(𝐴𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5411, 2, 53sylancl 693 . . . . 5 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5522, 54eqsstr3d 3624 . . . 4 (𝑅 ∈ TosetRel → ((𝐴𝐵) ∪ 𝐶) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5652, 55unssd 3772 . . 3 (𝑅 ∈ TosetRel → ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5749, 56eqssd 3605 . 2 (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴𝐵))) = ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
58 unass 3753 . 2 (({𝑋} ∪ (𝐴𝐵)) ∪ 𝐶) = ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
5957, 58syl6eqr 2678 1 (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴𝐵))) = (({𝑋} ∪ (𝐴𝐵)) ∪ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3o 1035   = wceq 1480  wcel 1992  wrex 2913  {crab 2916  Vcvv 3191  cun 3558  cin 3559  wss 3560  𝒫 cpw 4135  {csn 4153   cuni 4407   class class class wbr 4618  cmpt 4678  dom cdm 5079  ran crn 5080  cfv 5850  cmpt2 6607  ficfi 8261   TosetRel ctsr 17115
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  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 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-fin 7904  df-fi 8262  df-ps 17116  df-tsr 17117
This theorem is referenced by: (None)
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