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Theorem neiptopreu 22282
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 22259, innei 22274, elnei 22260 and neissex 22276, 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 22274 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 9148 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 22280 . . . 4 (𝜑𝐽 ∈ Top)
9 toptopon2 22065 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
108, 9sylib 217 . . 3 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
111, 2, 3, 4, 5, 6, 7neiptopuni 22279 . . . 4 (𝜑𝑋 = 𝐽)
1211fveq2d 6775 . . 3 (𝜑 → (TopOn‘𝑋) = (TopOn‘ 𝐽))
1310, 12eleqtrrd 2844 . 2 (𝜑𝐽 ∈ (TopOn‘𝑋))
141, 2, 3, 4, 5, 6, 7neiptopnei 22281 . 2 (𝜑𝑁 = (𝑝𝑋 ↦ ((nei‘𝐽)‘{𝑝})))
15 nfv 1921 . . . . . . . . . 10 𝑝(𝜑𝑗 ∈ (TopOn‘𝑋))
16 nfmpt1 5187 . . . . . . . . . . 11 𝑝(𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))
1716nfeq2 2926 . . . . . . . . . 10 𝑝 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))
1815, 17nfan 1906 . . . . . . . . 9 𝑝((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
19 nfv 1921 . . . . . . . . 9 𝑝 𝑏𝑋
2018, 19nfan 1906 . . . . . . . 8 𝑝(((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋)
21 simpllr 773 . . . . . . . . . . 11 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
22 simpr 485 . . . . . . . . . . . 12 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) → 𝑏𝑋)
2322sselda 3926 . . . . . . . . . . 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 6786 . . . . . . . . . . . 12 ((𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ∧ 𝑝𝑋) → ((nei‘𝑗)‘{𝑝}) ∈ V)
2624, 25fvmpt2d 6885 . . . . . . . . . . 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 2746 . . . . . . . . 9 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → ((nei‘𝑗)‘{𝑝}) = (𝑁𝑝))
2928eleq2d 2826 . . . . . . . 8 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → (𝑏 ∈ ((nei‘𝑗)‘{𝑝}) ↔ 𝑏 ∈ (𝑁𝑝)))
3020, 29ralbida 3159 . . . . . . 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 22074 . . . . . . . . 9 ((𝑗 ∈ (TopOn‘𝑋) ∧ 𝑏𝑗) → 𝑏𝑋)
3332ad4ant24 751 . . . . . . . 8 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑗) → 𝑏𝑋)
34 topontop 22060 . . . . . . . . . . 11 (𝑗 ∈ (TopOn‘𝑋) → 𝑗 ∈ Top)
3534ad2antlr 724 . . . . . . . . . 10 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → 𝑗 ∈ Top)
36 opnnei 22269 . . . . . . . . . 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 477 . . . . . . . 8 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑗) → ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))
3933, 38jca 512 . . . . . . 7 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑗) → (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})))
4037biimpar 478 . . . . . . . 8 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})) → 𝑏𝑗)
4140adantrl 713 . . . . . . 7 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))) → 𝑏𝑗)
4239, 41impbida 798 . . . . . 6 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏𝑗 ↔ (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))))
431neipeltop 22278 . . . . . . 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 2738 . . . 4 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → 𝑗 = 𝐽)
4746ex 413 . . 3 ((𝜑𝑗 ∈ (TopOn‘𝑋)) → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑗 = 𝐽))
4847ralrimiva 3110 . 2 (𝜑 → ∀𝑗 ∈ (TopOn‘𝑋)(𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑗 = 𝐽))
49 simpl 483 . . . . . . 7 ((𝑗 = 𝐽𝑝𝑋) → 𝑗 = 𝐽)
5049fveq2d 6775 . . . . . 6 ((𝑗 = 𝐽𝑝𝑋) → (nei‘𝑗) = (nei‘𝐽))
5150fveq1d 6773 . . . . 5 ((𝑗 = 𝐽𝑝𝑋) → ((nei‘𝑗)‘{𝑝}) = ((nei‘𝐽)‘{𝑝}))
5251mpteq2dva 5179 . . . 4 (𝑗 = 𝐽 → (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) = (𝑝𝑋 ↦ ((nei‘𝐽)‘{𝑝})))
5352eqeq2d 2751 . . 3 (𝑗 = 𝐽 → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝐽)‘{𝑝}))))
5453eqreu 3668 . 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 1370 1 (𝜑 → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1542  wcel 2110  wral 3066  wrex 3067  ∃!wreu 3068  {crab 3070  Vcvv 3431  wss 3892  𝒫 cpw 4539  {csn 4567   cuni 4845  cmpt 5162  wf 6428  cfv 6432  ficfi 9147  Topctop 22040  TopOnctopon 22057  neicnei 22246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-om 7707  df-1o 8288  df-en 8717  df-fin 8720  df-fi 9148  df-top 22041  df-topon 22058  df-ntr 22169  df-nei 22247
This theorem is referenced by:  ustuqtop  23396
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