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Theorem neiptopreu 23081
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 23058, innei 23073, elnei 23059 and neissex 23075, 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 23073 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 9436 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 23079 . . . 4 (𝜑𝐽 ∈ Top)
9 toptopon2 22864 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
108, 9sylib 217 . . 3 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
111, 2, 3, 4, 5, 6, 7neiptopuni 23078 . . . 4 (𝜑𝑋 = 𝐽)
1211fveq2d 6900 . . 3 (𝜑 → (TopOn‘𝑋) = (TopOn‘ 𝐽))
1310, 12eleqtrrd 2828 . 2 (𝜑𝐽 ∈ (TopOn‘𝑋))
141, 2, 3, 4, 5, 6, 7neiptopnei 23080 . 2 (𝜑𝑁 = (𝑝𝑋 ↦ ((nei‘𝐽)‘{𝑝})))
15 nfv 1909 . . . . . . . . . 10 𝑝(𝜑𝑗 ∈ (TopOn‘𝑋))
16 nfmpt1 5257 . . . . . . . . . . 11 𝑝(𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))
1716nfeq2 2909 . . . . . . . . . 10 𝑝 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))
1815, 17nfan 1894 . . . . . . . . 9 𝑝((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
19 nfv 1909 . . . . . . . . 9 𝑝 𝑏𝑋
2018, 19nfan 1894 . . . . . . . 8 𝑝(((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋)
21 simpllr 774 . . . . . . . . . . 11 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
22 simpr 483 . . . . . . . . . . . 12 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) → 𝑏𝑋)
2322sselda 3976 . . . . . . . . . . 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 6911 . . . . . . . . . . . 12 ((𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ∧ 𝑝𝑋) → ((nei‘𝑗)‘{𝑝}) ∈ V)
2624, 25fvmpt2d 7017 . . . . . . . . . . 11 ((𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ∧ 𝑝𝑋) → (𝑁𝑝) = ((nei‘𝑗)‘{𝑝}))
2721, 23, 26syl2anc 582 . . . . . . . . . 10 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → (𝑁𝑝) = ((nei‘𝑗)‘{𝑝}))
2827eqcomd 2731 . . . . . . . . 9 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → ((nei‘𝑗)‘{𝑝}) = (𝑁𝑝))
2928eleq2d 2811 . . . . . . . 8 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → (𝑏 ∈ ((nei‘𝑗)‘{𝑝}) ↔ 𝑏 ∈ (𝑁𝑝)))
3020, 29ralbida 3257 . . . . . . 7 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) → (∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}) ↔ ∀𝑝𝑏 𝑏 ∈ (𝑁𝑝)))
3130pm5.32da 577 . . . . . 6 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → ((𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})) ↔ (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ (𝑁𝑝))))
32 toponss 22873 . . . . . . . . 9 ((𝑗 ∈ (TopOn‘𝑋) ∧ 𝑏𝑗) → 𝑏𝑋)
3332ad4ant24 752 . . . . . . . 8 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑗) → 𝑏𝑋)
34 topontop 22859 . . . . . . . . . . 11 (𝑗 ∈ (TopOn‘𝑋) → 𝑗 ∈ Top)
3534ad2antlr 725 . . . . . . . . . 10 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → 𝑗 ∈ Top)
36 opnnei 23068 . . . . . . . . . 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 475 . . . . . . . 8 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑗) → ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))
3933, 38jca 510 . . . . . . 7 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑗) → (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})))
4037biimpar 476 . . . . . . . 8 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})) → 𝑏𝑗)
4140adantrl 714 . . . . . . 7 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))) → 𝑏𝑗)
4239, 41impbida 799 . . . . . 6 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏𝑗 ↔ (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))))
431neipeltop 23077 . . . . . . 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 310 . . . . 5 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏𝑗𝑏𝐽))
4645eqrdv 2723 . . . 4 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → 𝑗 = 𝐽)
4746ex 411 . . 3 ((𝜑𝑗 ∈ (TopOn‘𝑋)) → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑗 = 𝐽))
4847ralrimiva 3135 . 2 (𝜑 → ∀𝑗 ∈ (TopOn‘𝑋)(𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑗 = 𝐽))
49 simpl 481 . . . . . . 7 ((𝑗 = 𝐽𝑝𝑋) → 𝑗 = 𝐽)
5049fveq2d 6900 . . . . . 6 ((𝑗 = 𝐽𝑝𝑋) → (nei‘𝑗) = (nei‘𝐽))
5150fveq1d 6898 . . . . 5 ((𝑗 = 𝐽𝑝𝑋) → ((nei‘𝑗)‘{𝑝}) = ((nei‘𝐽)‘{𝑝}))
5251mpteq2dva 5249 . . . 4 (𝑗 = 𝐽 → (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) = (𝑝𝑋 ↦ ((nei‘𝐽)‘{𝑝})))
5352eqeq2d 2736 . . 3 (𝑗 = 𝐽 → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝐽)‘{𝑝}))))
5453eqreu 3721 . 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 1368 1 (𝜑 → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  wrex 3059  ∃!wreu 3361  {crab 3418  Vcvv 3461  wss 3944  𝒫 cpw 4604  {csn 4630   cuni 4909  cmpt 5232  wf 6545  cfv 6549  ficfi 9435  Topctop 22839  TopOnctopon 22856  neicnei 23045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-om 7872  df-1o 8487  df-en 8965  df-fin 8968  df-fi 9436  df-top 22840  df-topon 22857  df-ntr 22968  df-nei 23046
This theorem is referenced by:  ustuqtop  24195
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