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Theorem neiptopreu 20842
Description: If, to each element 𝑃 of a set 𝑋, we associate a set (𝑁𝑃) fulfilling the properties Vi, Vii, Viii and property Viv of [BourbakiTop1] p. I.2. , corresponding to ssnei 20819, innei 20834, elnei 20820 and neissex 20836, then there is a unique topology 𝑗 such that for any point 𝑝, (𝑁𝑝) is the set of neighborhoods of 𝑝. Proposition 2 of [BourbakiTop1] p. I.3. This can be used to build a topology from a set of neighborhoods. Note that the additional condition that 𝑋 is a neighborhood of all points was added. (Contributed by Thierry Arnoux, 6-Jan-2018.)
Hypotheses
Ref Expression
neiptop.o 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
neiptop.0 (𝜑𝑁:𝑋⟶𝒫 𝒫 𝑋)
neiptop.1 ((((𝜑𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))
neiptop.2 ((𝜑𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
neiptop.3 (((𝜑𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
neiptop.4 (((𝜑𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑞𝑏 𝑎 ∈ (𝑁𝑞))
neiptop.5 ((𝜑𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
Assertion
Ref Expression
neiptopreu (𝜑 → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
Distinct variable groups:   𝑝,𝑎,𝑁   𝑋,𝑎,𝑏,𝑝   𝐽,𝑎,𝑝   𝑋,𝑝   𝜑,𝑝   𝑁,𝑏   𝑋,𝑏   𝜑,𝑎,𝑏,𝑞,𝑝   𝑁,𝑝,𝑞   𝑋,𝑞   𝜑,𝑞   𝑗,𝑎,𝑏,𝐽,𝑝   𝑗,𝑞,𝑁   𝑗,𝑋   𝜑,𝑗
Allowed substitution hint:   𝐽(𝑞)

Proof of Theorem neiptopreu
StepHypRef Expression
1 neiptop.o . . . . 5 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
2 neiptop.0 . . . . 5 (𝜑𝑁:𝑋⟶𝒫 𝒫 𝑋)
3 neiptop.1 . . . . 5 ((((𝜑𝑝𝑋) ∧ 𝑎𝑏𝑏𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑏 ∈ (𝑁𝑝))
4 neiptop.2 . . . . 5 ((𝜑𝑝𝑋) → (fi‘(𝑁𝑝)) ⊆ (𝑁𝑝))
5 neiptop.3 . . . . 5 (((𝜑𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → 𝑝𝑎)
6 neiptop.4 . . . . 5 (((𝜑𝑝𝑋) ∧ 𝑎 ∈ (𝑁𝑝)) → ∃𝑏 ∈ (𝑁𝑝)∀𝑞𝑏 𝑎 ∈ (𝑁𝑞))
7 neiptop.5 . . . . 5 ((𝜑𝑝𝑋) → 𝑋 ∈ (𝑁𝑝))
81, 2, 3, 4, 5, 6, 7neiptoptop 20840 . . . 4 (𝜑𝐽 ∈ Top)
9 eqid 2626 . . . . 5 𝐽 = 𝐽
109toptopon 20643 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
118, 10sylib 208 . . 3 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
121, 2, 3, 4, 5, 6, 7neiptopuni 20839 . . . 4 (𝜑𝑋 = 𝐽)
1312fveq2d 6154 . . 3 (𝜑 → (TopOn‘𝑋) = (TopOn‘ 𝐽))
1411, 13eleqtrrd 2707 . 2 (𝜑𝐽 ∈ (TopOn‘𝑋))
151, 2, 3, 4, 5, 6, 7neiptopnei 20841 . 2 (𝜑𝑁 = (𝑝𝑋 ↦ ((nei‘𝐽)‘{𝑝})))
16 nfv 1845 . . . . . . . . . 10 𝑝(𝜑𝑗 ∈ (TopOn‘𝑋))
17 nfmpt1 4712 . . . . . . . . . . 11 𝑝(𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))
1817nfeq2 2782 . . . . . . . . . 10 𝑝 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))
1916, 18nfan 1830 . . . . . . . . 9 𝑝((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
20 nfv 1845 . . . . . . . . 9 𝑝 𝑏𝑋
2119, 20nfan 1830 . . . . . . . 8 𝑝(((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋)
22 simpllr 798 . . . . . . . . . . 11 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
23 simpr 477 . . . . . . . . . . . 12 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) → 𝑏𝑋)
2423sselda 3588 . . . . . . . . . . 11 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → 𝑝𝑋)
25 id 22 . . . . . . . . . . . 12 (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
26 fvex 6160 . . . . . . . . . . . . 13 ((nei‘𝑗)‘{𝑝}) ∈ V
2726a1i 11 . . . . . . . . . . . 12 ((𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ∧ 𝑝𝑋) → ((nei‘𝑗)‘{𝑝}) ∈ V)
2825, 27fvmpt2d 6251 . . . . . . . . . . 11 ((𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ∧ 𝑝𝑋) → (𝑁𝑝) = ((nei‘𝑗)‘{𝑝}))
2922, 24, 28syl2anc 692 . . . . . . . . . 10 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → (𝑁𝑝) = ((nei‘𝑗)‘{𝑝}))
3029eqcomd 2632 . . . . . . . . 9 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → ((nei‘𝑗)‘{𝑝}) = (𝑁𝑝))
3130eleq2d 2689 . . . . . . . 8 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → (𝑏 ∈ ((nei‘𝑗)‘{𝑝}) ↔ 𝑏 ∈ (𝑁𝑝)))
3221, 31ralbida 2981 . . . . . . 7 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) → (∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}) ↔ ∀𝑝𝑏 𝑏 ∈ (𝑁𝑝)))
3332pm5.32da 672 . . . . . 6 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → ((𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})) ↔ (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ (𝑁𝑝))))
34 simpllr 798 . . . . . . . . 9 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑗) → 𝑗 ∈ (TopOn‘𝑋))
35 simpr 477 . . . . . . . . 9 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑗) → 𝑏𝑗)
36 toponss 20639 . . . . . . . . 9 ((𝑗 ∈ (TopOn‘𝑋) ∧ 𝑏𝑗) → 𝑏𝑋)
3734, 35, 36syl2anc 692 . . . . . . . 8 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑗) → 𝑏𝑋)
38 topontop 20636 . . . . . . . . . . 11 (𝑗 ∈ (TopOn‘𝑋) → 𝑗 ∈ Top)
3938ad2antlr 762 . . . . . . . . . 10 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → 𝑗 ∈ Top)
40 opnnei 20829 . . . . . . . . . 10 (𝑗 ∈ Top → (𝑏𝑗 ↔ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})))
4139, 40syl 17 . . . . . . . . 9 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏𝑗 ↔ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})))
4241biimpa 501 . . . . . . . 8 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑗) → ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))
4337, 42jca 554 . . . . . . 7 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑗) → (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})))
4441biimpar 502 . . . . . . . 8 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})) → 𝑏𝑗)
4544adantrl 751 . . . . . . 7 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))) → 𝑏𝑗)
4643, 45impbida 876 . . . . . 6 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏𝑗 ↔ (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))))
471neipeltop 20838 . . . . . . 7 (𝑏𝐽 ↔ (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ (𝑁𝑝)))
4847a1i 11 . . . . . 6 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏𝐽 ↔ (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ (𝑁𝑝))))
4933, 46, 483bitr4d 300 . . . . 5 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏𝑗𝑏𝐽))
5049eqrdv 2624 . . . 4 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → 𝑗 = 𝐽)
5150ex 450 . . 3 ((𝜑𝑗 ∈ (TopOn‘𝑋)) → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑗 = 𝐽))
5251ralrimiva 2965 . 2 (𝜑 → ∀𝑗 ∈ (TopOn‘𝑋)(𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑗 = 𝐽))
53 simpl 473 . . . . . . 7 ((𝑗 = 𝐽𝑝𝑋) → 𝑗 = 𝐽)
5453fveq2d 6154 . . . . . 6 ((𝑗 = 𝐽𝑝𝑋) → (nei‘𝑗) = (nei‘𝐽))
5554fveq1d 6152 . . . . 5 ((𝑗 = 𝐽𝑝𝑋) → ((nei‘𝑗)‘{𝑝}) = ((nei‘𝐽)‘{𝑝}))
5655mpteq2dva 4709 . . . 4 (𝑗 = 𝐽 → (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) = (𝑝𝑋 ↦ ((nei‘𝐽)‘{𝑝})))
5756eqeq2d 2636 . . 3 (𝑗 = 𝐽 → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝐽)‘{𝑝}))))
5857eqreu 3385 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝐽)‘{𝑝})) ∧ ∀𝑗 ∈ (TopOn‘𝑋)(𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑗 = 𝐽)) → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
5914, 15, 52, 58syl3anc 1323 1 (𝜑 → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  wrex 2913  ∃!wreu 2914  {crab 2916  Vcvv 3191  wss 3560  𝒫 cpw 4135  {csn 4153   cuni 4407  cmpt 4678  wf 5846  cfv 5850  ficfi 8261  Topctop 20612  TopOnctopon 20613  neicnei 20806
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-rep 4736  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-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-top 20616  df-topon 20618  df-ntr 20729  df-nei 20807
This theorem is referenced by:  ustuqtop  21955
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