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Theorem neiptopreu 23141
Description: If, to each element 𝑃 of a set 𝑋, we associate a set (𝑁𝑃) fulfilling Properties Vi, Vii, Viii and Property Viv of [BourbakiTop1] p. I.2. , corresponding to ssnei 23118, innei 23133, elnei 23119 and neissex 23135, 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 innei 23133 uses binary intersections whereas Property Vii mentions finite intersections (which includes the empty intersection of subsets of 𝑋, which is equal to 𝑋), so we add the hypothesis that 𝑋 is a neighborhood of all points. TODO: when df-fi 9451 includes the empty intersection, remove that extra hypothesis. (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 23139 . . . 4 (𝜑𝐽 ∈ Top)
9 toptopon2 22924 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
108, 9sylib 218 . . 3 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
111, 2, 3, 4, 5, 6, 7neiptopuni 23138 . . . 4 (𝜑𝑋 = 𝐽)
1211fveq2d 6910 . . 3 (𝜑 → (TopOn‘𝑋) = (TopOn‘ 𝐽))
1310, 12eleqtrrd 2844 . 2 (𝜑𝐽 ∈ (TopOn‘𝑋))
141, 2, 3, 4, 5, 6, 7neiptopnei 23140 . 2 (𝜑𝑁 = (𝑝𝑋 ↦ ((nei‘𝐽)‘{𝑝})))
15 nfv 1914 . . . . . . . . . 10 𝑝(𝜑𝑗 ∈ (TopOn‘𝑋))
16 nfmpt1 5250 . . . . . . . . . . 11 𝑝(𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))
1716nfeq2 2923 . . . . . . . . . 10 𝑝 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))
1815, 17nfan 1899 . . . . . . . . 9 𝑝((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
19 nfv 1914 . . . . . . . . 9 𝑝 𝑏𝑋
2018, 19nfan 1899 . . . . . . . 8 𝑝(((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋)
21 simpllr 776 . . . . . . . . . . 11 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
22 simpr 484 . . . . . . . . . . . 12 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) → 𝑏𝑋)
2322sselda 3983 . . . . . . . . . . 11 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → 𝑝𝑋)
24 id 22 . . . . . . . . . . . 12 (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
25 fvexd 6921 . . . . . . . . . . . 12 ((𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ∧ 𝑝𝑋) → ((nei‘𝑗)‘{𝑝}) ∈ V)
2624, 25fvmpt2d 7029 . . . . . . . . . . 11 ((𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ∧ 𝑝𝑋) → (𝑁𝑝) = ((nei‘𝑗)‘{𝑝}))
2721, 23, 26syl2anc 584 . . . . . . . . . 10 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → (𝑁𝑝) = ((nei‘𝑗)‘{𝑝}))
2827eqcomd 2743 . . . . . . . . 9 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → ((nei‘𝑗)‘{𝑝}) = (𝑁𝑝))
2928eleq2d 2827 . . . . . . . 8 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → (𝑏 ∈ ((nei‘𝑗)‘{𝑝}) ↔ 𝑏 ∈ (𝑁𝑝)))
3020, 29ralbida 3270 . . . . . . 7 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) → (∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}) ↔ ∀𝑝𝑏 𝑏 ∈ (𝑁𝑝)))
3130pm5.32da 579 . . . . . 6 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → ((𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})) ↔ (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ (𝑁𝑝))))
32 toponss 22933 . . . . . . . . 9 ((𝑗 ∈ (TopOn‘𝑋) ∧ 𝑏𝑗) → 𝑏𝑋)
3332ad4ant24 754 . . . . . . . 8 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑗) → 𝑏𝑋)
34 topontop 22919 . . . . . . . . . . 11 (𝑗 ∈ (TopOn‘𝑋) → 𝑗 ∈ Top)
3534ad2antlr 727 . . . . . . . . . 10 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → 𝑗 ∈ Top)
36 opnnei 23128 . . . . . . . . . 10 (𝑗 ∈ Top → (𝑏𝑗 ↔ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})))
3735, 36syl 17 . . . . . . . . 9 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏𝑗 ↔ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})))
3837biimpa 476 . . . . . . . 8 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑗) → ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))
3933, 38jca 511 . . . . . . 7 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑗) → (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})))
4037biimpar 477 . . . . . . . 8 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})) → 𝑏𝑗)
4140adantrl 716 . . . . . . 7 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))) → 𝑏𝑗)
4239, 41impbida 801 . . . . . 6 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏𝑗 ↔ (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))))
431neipeltop 23137 . . . . . . 7 (𝑏𝐽 ↔ (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ (𝑁𝑝)))
4443a1i 11 . . . . . 6 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏𝐽 ↔ (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ (𝑁𝑝))))
4531, 42, 443bitr4d 311 . . . . 5 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏𝑗𝑏𝐽))
4645eqrdv 2735 . . . 4 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → 𝑗 = 𝐽)
4746ex 412 . . 3 ((𝜑𝑗 ∈ (TopOn‘𝑋)) → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑗 = 𝐽))
4847ralrimiva 3146 . 2 (𝜑 → ∀𝑗 ∈ (TopOn‘𝑋)(𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑗 = 𝐽))
49 simpl 482 . . . . . . 7 ((𝑗 = 𝐽𝑝𝑋) → 𝑗 = 𝐽)
5049fveq2d 6910 . . . . . 6 ((𝑗 = 𝐽𝑝𝑋) → (nei‘𝑗) = (nei‘𝐽))
5150fveq1d 6908 . . . . 5 ((𝑗 = 𝐽𝑝𝑋) → ((nei‘𝑗)‘{𝑝}) = ((nei‘𝐽)‘{𝑝}))
5251mpteq2dva 5242 . . . 4 (𝑗 = 𝐽 → (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) = (𝑝𝑋 ↦ ((nei‘𝐽)‘{𝑝})))
5352eqeq2d 2748 . . 3 (𝑗 = 𝐽 → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝐽)‘{𝑝}))))
5453eqreu 3735 . 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})))
5513, 14, 48, 54syl3anc 1373 1 (𝜑 → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  ∃!wreu 3378  {crab 3436  Vcvv 3480  wss 3951  𝒫 cpw 4600  {csn 4626   cuni 4907  cmpt 5225  wf 6557  cfv 6561  ficfi 9450  Topctop 22899  TopOnctopon 22916  neicnei 23105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-2o 8507  df-en 8986  df-fin 8989  df-fi 9451  df-top 22900  df-topon 22917  df-ntr 23028  df-nei 23106
This theorem is referenced by:  ustuqtop  24255
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