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Theorem neiptopreu 21736
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 21713, innei 21728, elnei 21714 and neissex 21730, 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 21728 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 8863 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 21734 . . . 4 (𝜑𝐽 ∈ Top)
9 toptopon2 21521 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
108, 9sylib 221 . . 3 (𝜑𝐽 ∈ (TopOn‘ 𝐽))
111, 2, 3, 4, 5, 6, 7neiptopuni 21733 . . . 4 (𝜑𝑋 = 𝐽)
1211fveq2d 6656 . . 3 (𝜑 → (TopOn‘𝑋) = (TopOn‘ 𝐽))
1310, 12eleqtrrd 2917 . 2 (𝜑𝐽 ∈ (TopOn‘𝑋))
141, 2, 3, 4, 5, 6, 7neiptopnei 21735 . 2 (𝜑𝑁 = (𝑝𝑋 ↦ ((nei‘𝐽)‘{𝑝})))
15 nfv 1915 . . . . . . . . . 10 𝑝(𝜑𝑗 ∈ (TopOn‘𝑋))
16 nfmpt1 5140 . . . . . . . . . . 11 𝑝(𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))
1716nfeq2 2996 . . . . . . . . . 10 𝑝 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))
1815, 17nfan 1900 . . . . . . . . 9 𝑝((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
19 nfv 1915 . . . . . . . . 9 𝑝 𝑏𝑋
2018, 19nfan 1900 . . . . . . . 8 𝑝(((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋)
21 simpllr 775 . . . . . . . . . . 11 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
22 simpr 488 . . . . . . . . . . . 12 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) → 𝑏𝑋)
2322sselda 3942 . . . . . . . . . . 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 6667 . . . . . . . . . . . 12 ((𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ∧ 𝑝𝑋) → ((nei‘𝑗)‘{𝑝}) ∈ V)
2624, 25fvmpt2d 6763 . . . . . . . . . . 11 ((𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ∧ 𝑝𝑋) → (𝑁𝑝) = ((nei‘𝑗)‘{𝑝}))
2721, 23, 26syl2anc 587 . . . . . . . . . 10 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → (𝑁𝑝) = ((nei‘𝑗)‘{𝑝}))
2827eqcomd 2828 . . . . . . . . 9 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → ((nei‘𝑗)‘{𝑝}) = (𝑁𝑝))
2928eleq2d 2899 . . . . . . . 8 (((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) ∧ 𝑝𝑏) → (𝑏 ∈ ((nei‘𝑗)‘{𝑝}) ↔ 𝑏 ∈ (𝑁𝑝)))
3020, 29ralbida 3219 . . . . . . 7 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑋) → (∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}) ↔ ∀𝑝𝑏 𝑏 ∈ (𝑁𝑝)))
3130pm5.32da 582 . . . . . 6 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → ((𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})) ↔ (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ (𝑁𝑝))))
32 toponss 21530 . . . . . . . . 9 ((𝑗 ∈ (TopOn‘𝑋) ∧ 𝑏𝑗) → 𝑏𝑋)
3332ad4ant24 753 . . . . . . . 8 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑗) → 𝑏𝑋)
34 topontop 21516 . . . . . . . . . . 11 (𝑗 ∈ (TopOn‘𝑋) → 𝑗 ∈ Top)
3534ad2antlr 726 . . . . . . . . . 10 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → 𝑗 ∈ Top)
36 opnnei 21723 . . . . . . . . . 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 480 . . . . . . . 8 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑗) → ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))
3933, 38jca 515 . . . . . . 7 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏𝑗) → (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})))
4037biimpar 481 . . . . . . . 8 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})) → 𝑏𝑗)
4140adantrl 715 . . . . . . 7 ((((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))) → 𝑏𝑗)
4239, 41impbida 800 . . . . . 6 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏𝑗 ↔ (𝑏𝑋 ∧ ∀𝑝𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))))
431neipeltop 21732 . . . . . . 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 314 . . . . 5 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏𝑗𝑏𝐽))
4645eqrdv 2820 . . . 4 (((𝜑𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → 𝑗 = 𝐽)
4746ex 416 . . 3 ((𝜑𝑗 ∈ (TopOn‘𝑋)) → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑗 = 𝐽))
4847ralrimiva 3174 . 2 (𝜑 → ∀𝑗 ∈ (TopOn‘𝑋)(𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑗 = 𝐽))
49 simpl 486 . . . . . . 7 ((𝑗 = 𝐽𝑝𝑋) → 𝑗 = 𝐽)
5049fveq2d 6656 . . . . . 6 ((𝑗 = 𝐽𝑝𝑋) → (nei‘𝑗) = (nei‘𝐽))
5150fveq1d 6654 . . . . 5 ((𝑗 = 𝐽𝑝𝑋) → ((nei‘𝑗)‘{𝑝}) = ((nei‘𝐽)‘{𝑝}))
5251mpteq2dva 5137 . . . 4 (𝑗 = 𝐽 → (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) = (𝑝𝑋 ↦ ((nei‘𝐽)‘{𝑝})))
5352eqeq2d 2833 . . 3 (𝑗 = 𝐽 → (𝑁 = (𝑝𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ 𝑁 = (𝑝𝑋 ↦ ((nei‘𝐽)‘{𝑝}))))
5453eqreu 3695 . 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wral 3130  wrex 3131  ∃!wreu 3132  {crab 3134  Vcvv 3469  wss 3908  𝒫 cpw 4511  {csn 4539   cuni 4813  cmpt 5122  wf 6330  cfv 6334  ficfi 8862  Topctop 21496  TopOnctopon 21513  neicnei 21700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-fin 8500  df-fi 8863  df-top 21497  df-topon 21514  df-ntr 21623  df-nei 21701
This theorem is referenced by:  ustuqtop  22850
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