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Theorem ordtbas 21276
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 5064 . . . . . 6 {𝑋} ∈ V
2 ssun2 3939 . . . . . . 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 21274 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 = ({𝑋} ∪ (𝐴𝐵)))
7 dmexg 7295 . . . . . . . . . 10 (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
83, 7syl5eqel 2848 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 ∈ V)
96, 8eqeltrrd 2845 . . . . . . . 8 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
10 uniexb 7171 . . . . . . . 8 (({𝑋} ∪ (𝐴𝐵)) ∈ V ↔ ({𝑋} ∪ (𝐴𝐵)) ∈ V)
119, 10sylibr 225 . . . . . . 7 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ∈ V)
12 ssexg 4965 . . . . . . 7 (((𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵)) ∧ ({𝑋} ∪ (𝐴𝐵)) ∈ V) → (𝐴𝐵) ∈ V)
132, 11, 12sylancr 581 . . . . . 6 (𝑅 ∈ TosetRel → (𝐴𝐵) ∈ V)
14 elfiun 8543 . . . . . 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 581 . . . . 5 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘({𝑋} ∪ (𝐴𝐵))) ↔ (𝑧 ∈ (fi‘{𝑋}) ∨ 𝑧 ∈ (fi‘(𝐴𝐵)) ∨ ∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴𝐵))𝑧 = (𝑚𝑛))))
16 fisn 8540 . . . . . . . . 9 (fi‘{𝑋}) = {𝑋}
17 ssun1 3938 . . . . . . . . 9 {𝑋} ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
1816, 17eqsstri 3795 . . . . . . . 8 (fi‘{𝑋}) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
1918sseli 3757 . . . . . . 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 21275 . . . . . . . 8 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) = ((𝐴𝐵) ∪ 𝐶))
23 ssun2 3939 . . . . . . . 8 ((𝐴𝐵) ∪ 𝐶) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
2422, 23syl6eqss 3815 . . . . . . 7 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
2524sseld 3760 . . . . . 6 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘(𝐴𝐵)) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
26 fipwuni 8539 . . . . . . . . . . . . . . 15 (fi‘(𝐴𝐵)) ⊆ 𝒫 (𝐴𝐵)
2726sseli 3757 . . . . . . . . . . . . . 14 (𝑛 ∈ (fi‘(𝐴𝐵)) → 𝑛 ∈ 𝒫 (𝐴𝐵))
2827elpwid 4327 . . . . . . . . . . . . 13 (𝑛 ∈ (fi‘(𝐴𝐵)) → 𝑛 (𝐴𝐵))
2928ad2antll 720 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛 (𝐴𝐵))
302unissi 4619 . . . . . . . . . . . . . 14 (𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵))
3130, 6syl5sseqr 3814 . . . . . . . . . . . . 13 (𝑅 ∈ TosetRel → (𝐴𝐵) ⊆ 𝑋)
3231adantr 472 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝐴𝐵) ⊆ 𝑋)
3329, 32sstrd 3771 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛𝑋)
34 simprl 787 . . . . . . . . . . . . 13 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑚 ∈ (fi‘{𝑋}))
3534, 16syl6eleq 2854 . . . . . . . . . . . 12 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑚 ∈ {𝑋})
36 elsni 4351 . . . . . . . . . . . 12 (𝑚 ∈ {𝑋} → 𝑚 = 𝑋)
3735, 36syl 17 . . . . . . . . . . 11 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑚 = 𝑋)
3833, 37sseqtr4d 3802 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛𝑚)
39 sseqin2 3979 . . . . . . . . . 10 (𝑛𝑚 ↔ (𝑚𝑛) = 𝑛)
4038, 39sylib 209 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝑚𝑛) = 𝑛)
4124sselda 3761 . . . . . . . . . 10 ((𝑅 ∈ TosetRel ∧ 𝑛 ∈ (fi‘(𝐴𝐵))) → 𝑛 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
4241adantrl 707 . . . . . . . . 9 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → 𝑛 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
4340, 42eqeltrd 2844 . . . . . . . 8 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝑚𝑛) ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
44 eleq1 2832 . . . . . . . 8 (𝑧 = (𝑚𝑛) → (𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)) ↔ (𝑚𝑛) ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4543, 44syl5ibrcom 238 . . . . . . 7 ((𝑅 ∈ TosetRel ∧ (𝑚 ∈ (fi‘{𝑋}) ∧ 𝑛 ∈ (fi‘(𝐴𝐵)))) → (𝑧 = (𝑚𝑛) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4645rexlimdvva 3185 . . . . . 6 (𝑅 ∈ TosetRel → (∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴𝐵))𝑧 = (𝑚𝑛) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4720, 25, 463jaod 1553 . . . . 5 (𝑅 ∈ TosetRel → ((𝑧 ∈ (fi‘{𝑋}) ∨ 𝑧 ∈ (fi‘(𝐴𝐵)) ∨ ∃𝑚 ∈ (fi‘{𝑋})∃𝑛 ∈ (fi‘(𝐴𝐵))𝑧 = (𝑚𝑛)) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4815, 47sylbid 231 . . . 4 (𝑅 ∈ TosetRel → (𝑧 ∈ (fi‘({𝑋} ∪ (𝐴𝐵))) → 𝑧 ∈ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))))
4948ssrdv 3767 . . 3 (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴𝐵))) ⊆ ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
50 ssfii 8532 . . . . . 6 (({𝑋} ∪ (𝐴𝐵)) ∈ V → ({𝑋} ∪ (𝐴𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5111, 50syl 17 . . . . 5 (𝑅 ∈ TosetRel → ({𝑋} ∪ (𝐴𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5251unssad 3952 . . . 4 (𝑅 ∈ TosetRel → {𝑋} ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
53 fiss 8537 . . . . . 6 ((({𝑋} ∪ (𝐴𝐵)) ∈ V ∧ (𝐴𝐵) ⊆ ({𝑋} ∪ (𝐴𝐵))) → (fi‘(𝐴𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5411, 2, 53sylancl 580 . . . . 5 (𝑅 ∈ TosetRel → (fi‘(𝐴𝐵)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5522, 54eqsstr3d 3800 . . . 4 (𝑅 ∈ TosetRel → ((𝐴𝐵) ∪ 𝐶) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5652, 55unssd 3951 . . 3 (𝑅 ∈ TosetRel → ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)) ⊆ (fi‘({𝑋} ∪ (𝐴𝐵))))
5749, 56eqssd 3778 . 2 (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴𝐵))) = ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶)))
58 unass 3932 . 2 (({𝑋} ∪ (𝐴𝐵)) ∪ 𝐶) = ({𝑋} ∪ ((𝐴𝐵) ∪ 𝐶))
5957, 58syl6eqr 2817 1 (𝑅 ∈ TosetRel → (fi‘({𝑋} ∪ (𝐴𝐵))) = (({𝑋} ∪ (𝐴𝐵)) ∪ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3o 1106   = wceq 1652  wcel 2155  wrex 3056  {crab 3059  Vcvv 3350  cun 3730  cin 3731  wss 3732  𝒫 cpw 4315  {csn 4334   cuni 4594   class class class wbr 4809  cmpt 4888  dom cdm 5277  ran crn 5278  cfv 6068  cmpt2 6844  ficfi 8523   TosetRel ctsr 17465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-fin 8164  df-fi 8524  df-ps 17466  df-tsr 17467
This theorem is referenced by: (None)
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