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Theorem neissex 23135
Description: For any neighborhood 𝑁 of 𝑆, there is a neighborhood 𝑥 of 𝑆 such that 𝑁 is a neighborhood of all subsets of 𝑥. Generalization to subsets of Property Viv of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)
Assertion
Ref Expression
neissex ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦(𝑦𝑥𝑁 ∈ ((nei‘𝐽)‘𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑁,𝑦   𝑥,𝑆,𝑦

Proof of Theorem neissex
StepHypRef Expression
1 neii2 23116 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥𝐽 (𝑆𝑥𝑥𝑁))
2 opnneiss 23126 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑆𝑥) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
323expb 1121 . . . 4 ((𝐽 ∈ Top ∧ (𝑥𝐽𝑆𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
43adantrrr 725 . . 3 ((𝐽 ∈ Top ∧ (𝑥𝐽 ∧ (𝑆𝑥𝑥𝑁))) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
54adantlr 715 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽 ∧ (𝑆𝑥𝑥𝑁))) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
6 simplll 775 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) ∧ 𝑦𝑥) → 𝐽 ∈ Top)
7 simpll 767 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) → 𝐽 ∈ Top)
8 simpr 484 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) → 𝑥𝐽)
9 eqid 2737 . . . . . . . . . . . 12 𝐽 = 𝐽
109neii1 23114 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 𝐽)
1110adantr 480 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) → 𝑁 𝐽)
129opnssneib 23123 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑁 𝐽) → (𝑥𝑁𝑁 ∈ ((nei‘𝐽)‘𝑥)))
137, 8, 11, 12syl3anc 1373 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) → (𝑥𝑁𝑁 ∈ ((nei‘𝐽)‘𝑥)))
1413biimpa 476 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) ∧ 𝑥𝑁) → 𝑁 ∈ ((nei‘𝐽)‘𝑥))
1514anasss 466 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) → 𝑁 ∈ ((nei‘𝐽)‘𝑥))
1615adantr 480 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) ∧ 𝑦𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑥))
17 simpr 484 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) ∧ 𝑦𝑥) → 𝑦𝑥)
18 neiss 23117 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑥) ∧ 𝑦𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑦))
196, 16, 17, 18syl3anc 1373 . . . . 5 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) ∧ 𝑦𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑦))
2019ex 412 . . . 4 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) → (𝑦𝑥𝑁 ∈ ((nei‘𝐽)‘𝑦)))
2120adantrrl 724 . . 3 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽 ∧ (𝑆𝑥𝑥𝑁))) → (𝑦𝑥𝑁 ∈ ((nei‘𝐽)‘𝑦)))
2221alrimiv 1927 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽 ∧ (𝑆𝑥𝑥𝑁))) → ∀𝑦(𝑦𝑥𝑁 ∈ ((nei‘𝐽)‘𝑦)))
231, 5, 22reximssdv 3173 1 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥 ∈ ((nei‘𝐽)‘𝑆)∀𝑦(𝑦𝑥𝑁 ∈ ((nei‘𝐽)‘𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538  wcel 2108  wrex 3070  wss 3951   cuni 4907  cfv 6561  Topctop 22899  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
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  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-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-nei 23106
This theorem is referenced by: (None)
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