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Theorem ordtbas 23017
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 5421 . . . . . 6 {𝑋} ∈ V
2 ssun2 4165 . . . . . . 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 23015 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 = ({𝑋} ∪ (𝐴𝐵)))
7 dmexg 7887 . . . . . . . . . 10 (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
83, 7eqeltrid 2829 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 ∈ V)
96, 8eqeltrrd 2826 . . . . . . . 8 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
10 uniexb 7744 . . . . . . . 8 (({𝑋} ∪ (𝐴𝐵)) ∈ V ↔ ({𝑋} ∪ (𝐴𝐵)) ∈ V)
119, 10sylibr 233 . . . . . . 7 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
12 ssexg 5313 . . . . . . 7 (((𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵)) ∧ ({𝑋} ∪ (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
132, 11, 12sylancr 586 . . . . . 6 (𝑅 ∈ TosetRel → (𝐴𝐵) ∈ V)
14 elfiun 9420 . . . . . 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 586 . . . . 5 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘({𝑋} ∪ (𝐴𝐵))) ↔ (𝑧 ∈ (fi‘{𝑋}) ∨ 𝑧 ∈ (fi‘(𝐴𝐵)) ∨ ∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴𝐵))𝑧 = (𝑚𝑛))))
16 fisn 9417 . . . . . . . . 9 (fi‘{𝑋}) = {𝑋}
17 ssun1 4164 . . . . . . . . 9 {𝑋} ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
1816, 17eqsstri 4008 . . . . . . . 8 (fi‘{𝑋}) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
1918sseli 3970 . . . . . . 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 23016 . . . . . . . 8 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) = ((𝐴𝐵) ∪ 𝐶))
23 ssun2 4165 . . . . . . . 8 ((𝐴𝐵) ∪ 𝐶) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
2422, 23eqsstrdi 4028 . . . . . . 7 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
2524sseld 3973 . . . . . 6 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴𝐵)) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
26 fipwuni 9416 . . . . . . . . . . . . . . 15 (fi‘(𝐴𝐵)) ⊆ 𝒫 (𝐴𝐵)
2726sseli 3970 . . . . . . . . . . . . . 14 (𝑛 ∈ (fi‘(𝐴𝐵)) → 𝑛 ∈ 𝒫 (𝐴𝐵))
2827elpwid 4603 . . . . . . . . . . . . 13 (𝑛 ∈ (fi‘(𝐴𝐵)) → 𝑛 (𝐴𝐵))
2928ad2antll 726 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛 (𝐴𝐵))
302unissi 4908 . . . . . . . . . . . . . 14 (𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵))
3130, 6sseqtrrid 4027 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → (𝐴𝐵) ⊆ 𝑋)
3231adantr 480 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝐴𝐵) ⊆ 𝑋)
3329, 32sstrd 3984 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛𝑋)
34 simprl 768 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑚 ∈ (fi‘{𝑋}))
3534, 16eleqtrdi 2835 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑚 ∈ {𝑋})
36 elsni 4637 . . . . . . . . . . . 12 (𝑚 ∈ {𝑋} → 𝑚 = 𝑋)
3735, 36syl 17 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑚 = 𝑋)
3833, 37sseqtrrd 4015 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛𝑚)
39 sseqin2 4207 . . . . . . . . . 10 (𝑛𝑚 ↔ (𝑚𝑛) = 𝑛)
4038, 39sylib 217 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝑚𝑛) = 𝑛)
4124sselda 3974 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ 𝑛 ∈ (fi‘(𝐴𝐵))) → 𝑛 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
4241adantrl 713 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
4340, 42eqeltrd 2825 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝑚𝑛) ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
44 eleq1 2813 . . . . . . . 8 (𝑧 = (𝑚𝑛) → (𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)) ↔ (𝑚𝑛) ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4543, 44syl5ibrcom 246 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝑧 = (𝑚𝑛) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4645rexlimdvva 3203 . . . . . 6 (𝑅 ∈ TosetRel → (∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴𝐵))𝑧 = (𝑚𝑛) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4720, 25, 463jaod 1425 . . . . 5 (𝑅 ∈ TosetRel → ((𝑧 ∈ (fi‘{𝑋}) ∨ 𝑧 ∈ (fi‘(𝐴𝐵)) ∨ ∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴𝐵))𝑧 = (𝑚𝑛)) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4815, 47sylbid 239 . . . 4 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘({𝑋} ∪ (𝐴𝐵))) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4948ssrdv 3980 . . 3 (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴𝐵))) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
50 ssfii 9409 . . . . . 6 (({𝑋} ∪ (𝐴𝐵)) ∈ V → ({𝑋} ∪ (𝐴𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5111, 50syl 17 . . . . 5 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5251unssad 4179 . . . 4 (𝑅 ∈ TosetRel → {𝑋} ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
53 fiss 9414 . . . . . 6 ((({𝑋} ∪ (𝐴𝐵)) ∈ V ∧ (𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵))) → (fi‘(𝐴𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5411, 2, 53sylancl 585 . . . . 5 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5522, 54eqsstrrd 4013 . . . 4 (𝑅 ∈ TosetRel → ((𝐴𝐵) ∪ 𝐶) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5652, 55unssd 4178 . . 3 (𝑅 ∈ TosetRel → ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5749, 56eqssd 3991 . 2 (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴𝐵))) = ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
58 unass 4158 . 2 (({𝑋} ∪ (𝐴𝐵)) ∪ 𝐶) = ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
5957, 58eqtr4di 2782 1 (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴𝐵))) = (({𝑋} ∪ (𝐴𝐵)) ∪ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3o 1083   = wceq 1533  wcel 2098  wrex 3062  {crab 3424  Vcvv 3466  cun 3938  cin 3939  wss 3940  𝒫 cpw 4594  {csn 4620   cuni 4899   class class class wbr 5138  cmpt 5221  dom cdm 5666  ran crn 5667  cfv 6533  cmpo 7403  ficfi 9400   TosetRel ctsr 18519
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 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-1o 8461  df-er 8698  df-en 8935  df-fin 8938  df-fi 9401  df-ps 18520  df-tsr 18521
This theorem is referenced by: (None)
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