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Theorem neissex 22462
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 22443 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥𝐽 (𝑆𝑥𝑥𝑁))
2 opnneiss 22453 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑆𝑥) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
323expb 1120 . . . 4 ((𝐽 ∈ Top ∧ (𝑥𝐽𝑆𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
43adantrrr 723 . . 3 ((𝐽 ∈ Top ∧ (𝑥𝐽 ∧ (𝑆𝑥𝑥𝑁))) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
54adantlr 713 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽 ∧ (𝑆𝑥𝑥𝑁))) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
6 simplll 773 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) ∧ 𝑦𝑥) → 𝐽 ∈ Top)
7 simpll 765 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) → 𝐽 ∈ Top)
8 simpr 485 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) → 𝑥𝐽)
9 eqid 2736 . . . . . . . . . . . 12 𝐽 = 𝐽
109neii1 22441 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 𝐽)
1110adantr 481 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) → 𝑁 𝐽)
129opnssneib 22450 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑁 𝐽) → (𝑥𝑁𝑁 ∈ ((nei‘𝐽)‘𝑥)))
137, 8, 11, 12syl3anc 1371 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) → (𝑥𝑁𝑁 ∈ ((nei‘𝐽)‘𝑥)))
1413biimpa 477 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) ∧ 𝑥𝑁) → 𝑁 ∈ ((nei‘𝐽)‘𝑥))
1514anasss 467 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) → 𝑁 ∈ ((nei‘𝐽)‘𝑥))
1615adantr 481 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) ∧ 𝑦𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑥))
17 simpr 485 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) ∧ 𝑦𝑥) → 𝑦𝑥)
18 neiss 22444 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑥) ∧ 𝑦𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑦))
196, 16, 17, 18syl3anc 1371 . . . . 5 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) ∧ 𝑦𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑦))
2019ex 413 . . . 4 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) → (𝑦𝑥𝑁 ∈ ((nei‘𝐽)‘𝑦)))
2120adantrrl 722 . . 3 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽 ∧ (𝑆𝑥𝑥𝑁))) → (𝑦𝑥𝑁 ∈ ((nei‘𝐽)‘𝑦)))
2221alrimiv 1930 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽 ∧ (𝑆𝑥𝑥𝑁))) → ∀𝑦(𝑦𝑥𝑁 ∈ ((nei‘𝐽)‘𝑦)))
231, 5, 22reximssdv 3167 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 205  wa 396  wal 1539  wcel 2106  wrex 3071  wss 3908   cuni 4863  cfv 6493  Topctop 22226  neicnei 22432
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-top 22227  df-nei 22433
This theorem is referenced by: (None)
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