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Theorem ordtbas 23216
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 5442 . . . . . 6 {𝑋} ∈ V
2 ssun2 4189 . . . . . . 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 23214 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 = ({𝑋} ∪ (𝐴𝐵)))
7 dmexg 7924 . . . . . . . . . 10 (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
83, 7eqeltrid 2843 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 ∈ V)
96, 8eqeltrrd 2840 . . . . . . . 8 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
10 uniexb 7783 . . . . . . . 8 (({𝑋} ∪ (𝐴𝐵)) ∈ V ↔ ({𝑋} ∪ (𝐴𝐵)) ∈ V)
119, 10sylibr 234 . . . . . . 7 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
12 ssexg 5329 . . . . . . 7 (((𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵)) ∧ ({𝑋} ∪ (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
132, 11, 12sylancr 587 . . . . . 6 (𝑅 ∈ TosetRel → (𝐴𝐵) ∈ V)
14 elfiun 9468 . . . . . 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 587 . . . . 5 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘({𝑋} ∪ (𝐴𝐵))) ↔ (𝑧 ∈ (fi‘{𝑋}) ∨ 𝑧 ∈ (fi‘(𝐴𝐵)) ∨ ∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴𝐵))𝑧 = (𝑚𝑛))))
16 fisn 9465 . . . . . . . . 9 (fi‘{𝑋}) = {𝑋}
17 ssun1 4188 . . . . . . . . 9 {𝑋} ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
1816, 17eqsstri 4030 . . . . . . . 8 (fi‘{𝑋}) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
1918sseli 3991 . . . . . . 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 23215 . . . . . . . 8 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) = ((𝐴𝐵) ∪ 𝐶))
23 ssun2 4189 . . . . . . . 8 ((𝐴𝐵) ∪ 𝐶) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
2422, 23eqsstrdi 4050 . . . . . . 7 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
2524sseld 3994 . . . . . 6 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴𝐵)) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
26 fipwuni 9464 . . . . . . . . . . . . . . 15 (fi‘(𝐴𝐵)) ⊆ 𝒫 (𝐴𝐵)
2726sseli 3991 . . . . . . . . . . . . . 14 (𝑛 ∈ (fi‘(𝐴𝐵)) → 𝑛 ∈ 𝒫 (𝐴𝐵))
2827elpwid 4614 . . . . . . . . . . . . 13 (𝑛 ∈ (fi‘(𝐴𝐵)) → 𝑛 (𝐴𝐵))
2928ad2antll 729 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛 (𝐴𝐵))
302unissi 4921 . . . . . . . . . . . . . 14 (𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵))
3130, 6sseqtrrid 4049 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → (𝐴𝐵) ⊆ 𝑋)
3231adantr 480 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝐴𝐵) ⊆ 𝑋)
3329, 32sstrd 4006 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛𝑋)
34 simprl 771 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑚 ∈ (fi‘{𝑋}))
3534, 16eleqtrdi 2849 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑚 ∈ {𝑋})
36 elsni 4648 . . . . . . . . . . . 12 (𝑚 ∈ {𝑋} → 𝑚 = 𝑋)
3735, 36syl 17 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑚 = 𝑋)
3833, 37sseqtrrd 4037 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛𝑚)
39 sseqin2 4231 . . . . . . . . . 10 (𝑛𝑚 ↔ (𝑚𝑛) = 𝑛)
4038, 39sylib 218 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝑚𝑛) = 𝑛)
4124sselda 3995 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ 𝑛 ∈ (fi‘(𝐴𝐵))) → 𝑛 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
4241adantrl 716 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
4340, 42eqeltrd 2839 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝑚𝑛) ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
44 eleq1 2827 . . . . . . . 8 (𝑧 = (𝑚𝑛) → (𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)) ↔ (𝑚𝑛) ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4543, 44syl5ibrcom 247 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝑧 = (𝑚𝑛) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4645rexlimdvva 3211 . . . . . 6 (𝑅 ∈ TosetRel → (∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴𝐵))𝑧 = (𝑚𝑛) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4720, 25, 463jaod 1428 . . . . 5 (𝑅 ∈ TosetRel → ((𝑧 ∈ (fi‘{𝑋}) ∨ 𝑧 ∈ (fi‘(𝐴𝐵)) ∨ ∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴𝐵))𝑧 = (𝑚𝑛)) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4815, 47sylbid 240 . . . 4 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘({𝑋} ∪ (𝐴𝐵))) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4948ssrdv 4001 . . 3 (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴𝐵))) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
50 ssfii 9457 . . . . . 6 (({𝑋} ∪ (𝐴𝐵)) ∈ V → ({𝑋} ∪ (𝐴𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5111, 50syl 17 . . . . 5 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5251unssad 4203 . . . 4 (𝑅 ∈ TosetRel → {𝑋} ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
53 fiss 9462 . . . . . 6 ((({𝑋} ∪ (𝐴𝐵)) ∈ V ∧ (𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵))) → (fi‘(𝐴𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5411, 2, 53sylancl 586 . . . . 5 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5522, 54eqsstrrd 4035 . . . 4 (𝑅 ∈ TosetRel → ((𝐴𝐵) ∪ 𝐶) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5652, 55unssd 4202 . . 3 (𝑅 ∈ TosetRel → ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5749, 56eqssd 4013 . 2 (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴𝐵))) = ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
58 unass 4182 . 2 (({𝑋} ∪ (𝐴𝐵)) ∪ 𝐶) = ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
5957, 58eqtr4di 2793 1 (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴𝐵))) = (({𝑋} ∪ (𝐴𝐵)) ∪ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3o 1085   = wceq 1537  wcel 2106  wrex 3068  {crab 3433  Vcvv 3478  cun 3961  cin 3962  wss 3963  𝒫 cpw 4605  {csn 4631   cuni 4912   class class class wbr 5148  cmpt 5231  dom cdm 5689  ran crn 5690  cfv 6563  cmpo 7433  ficfi 9448   TosetRel ctsr 18623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-en 8985  df-fin 8988  df-fi 9449  df-ps 18624  df-tsr 18625
This theorem is referenced by: (None)
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