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Theorem ordtbas 20806
 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 4835 . . . . . 6 {𝑋} ∈ V
2 ssun2 3739 . . . . . . 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 20804 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 = ({𝑋} ∪ (𝐴𝐵)))
7 dmexg 6989 . . . . . . . . . 10 (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
83, 7syl5eqel 2692 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 ∈ V)
96, 8eqeltrrd 2689 . . . . . . . 8 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
10 uniexb 6866 . . . . . . . 8 (({𝑋} ∪ (𝐴𝐵)) ∈ V ↔ ({𝑋} ∪ (𝐴𝐵)) ∈ V)
119, 10sylibr 223 . . . . . . 7 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
12 ssexg 4732 . . . . . . 7 (((𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵)) ∧ ({𝑋} ∪ (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
132, 11, 12sylancr 694 . . . . . 6 (𝑅 ∈ TosetRel → (𝐴𝐵) ∈ V)
14 elfiun 8219 . . . . . 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 8216 . . . . . . . . 9 (fi‘{𝑋}) = {𝑋}
17 ssun1 3738 . . . . . . . . 9 {𝑋} ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
1816, 17eqsstri 3598 . . . . . . . 8 (fi‘{𝑋}) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
1918sseli 3564 . . . . . . 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 20805 . . . . . . . 8 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) = ((𝐴𝐵) ∪ 𝐶))
23 ssun2 3739 . . . . . . . 8 ((𝐴𝐵) ∪ 𝐶) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
2422, 23syl6eqss 3618 . . . . . . 7 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
2524sseld 3567 . . . . . 6 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴𝐵)) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
26 fipwuni 8215 . . . . . . . . . . . . . . 15 (fi‘(𝐴𝐵)) ⊆ 𝒫 (𝐴𝐵)
2726sseli 3564 . . . . . . . . . . . . . 14 (𝑛 ∈ (fi‘(𝐴𝐵)) → 𝑛 ∈ 𝒫 (𝐴𝐵))
2827elpwid 4118 . . . . . . . . . . . . 13 (𝑛 ∈ (fi‘(𝐴𝐵)) → 𝑛 (𝐴𝐵))
2928ad2antll 761 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛 (𝐴𝐵))
302unissi 4397 . . . . . . . . . . . . . 14 (𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵))
3130, 6syl5sseqr 3617 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → (𝐴𝐵) ⊆ 𝑋)
3231adantr 480 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝐴𝐵) ⊆ 𝑋)
3329, 32sstrd 3578 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛𝑋)
34 simprl 790 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑚 ∈ (fi‘{𝑋}))
3534, 16syl6eleq 2698 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑚 ∈ {𝑋})
36 elsni 4142 . . . . . . . . . . . 12 (𝑚 ∈ {𝑋} → 𝑚 = 𝑋)
3735, 36syl 17 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑚 = 𝑋)
3833, 37sseqtr4d 3605 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛𝑚)
39 sseqin2 3779 . . . . . . . . . 10 (𝑛𝑚 ↔ (𝑚𝑛) = 𝑛)
4038, 39sylib 207 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝑚𝑛) = 𝑛)
4124sselda 3568 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ 𝑛 ∈ (fi‘(𝐴𝐵))) → 𝑛 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
4241adantrl 748 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
4340, 42eqeltrd 2688 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝑚𝑛) ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
44 eleq1 2676 . . . . . . . 8 (𝑧 = (𝑚𝑛) → (𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)) ↔ (𝑚𝑛) ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4543, 44syl5ibrcom 236 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝑧 = (𝑚𝑛) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4645rexlimdvva 3020 . . . . . 6 (𝑅 ∈ TosetRel → (∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴𝐵))𝑧 = (𝑚𝑛) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4720, 25, 463jaod 1384 . . . . 5 (𝑅 ∈ TosetRel → ((𝑧 ∈ (fi‘{𝑋}) ∨ 𝑧 ∈ (fi‘(𝐴𝐵)) ∨ ∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴𝐵))𝑧 = (𝑚𝑛)) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4815, 47sylbid 229 . . . 4 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘({𝑋} ∪ (𝐴𝐵))) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4948ssrdv 3574 . . 3 (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴𝐵))) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
50 ssfii 8208 . . . . . 6 (({𝑋} ∪ (𝐴𝐵)) ∈ V → ({𝑋} ∪ (𝐴𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5111, 50syl 17 . . . . 5 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5251unssad 3752 . . . 4 (𝑅 ∈ TosetRel → {𝑋} ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
53 fiss 8213 . . . . . 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 3603 . . . 4 (𝑅 ∈ TosetRel → ((𝐴𝐵) ∪ 𝐶) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5652, 55unssd 3751 . . 3 (𝑅 ∈ TosetRel → ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5749, 56eqssd 3585 . 2 (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴𝐵))) = ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
58 unass 3732 . 2 (({𝑋} ∪ (𝐴𝐵)) ∪ 𝐶) = ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
5957, 58syl6eqr 2662 1 (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴𝐵))) = (({𝑋} ∪ (𝐴𝐵)) ∪ 𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ran crn 5039  ‘cfv 5804   ↦ cmpt2 6551  ficfi 8199   TosetRel ctsr 17022 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-ps 17023  df-tsr 17024 This theorem is referenced by: (None)
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