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Theorem neissex 23150
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 23131 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥𝐽 (𝑆𝑥𝑥𝑁))
2 opnneiss 23141 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑆𝑥) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
323expb 1119 . . . 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 2734 . . . . . . . . . . . 12 𝐽 = 𝐽
109neii1 23129 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 𝐽)
1110adantr 480 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) → 𝑁 𝐽)
129opnssneib 23138 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑁 𝐽) → (𝑥𝑁𝑁 ∈ ((nei‘𝐽)‘𝑥)))
137, 8, 11, 12syl3anc 1370 . . . . . . . . 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 23132 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑥) ∧ 𝑦𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑦))
196, 16, 17, 18syl3anc 1370 . . . . 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 1924 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽 ∧ (𝑆𝑥𝑥𝑁))) → ∀𝑦(𝑦𝑥𝑁 ∈ ((nei‘𝐽)‘𝑦)))
231, 5, 22reximssdv 3170 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 1534  wcel 2105  wrex 3067  wss 3962   cuni 4911  cfv 6562  Topctop 22914  neicnei 23120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-top 22915  df-nei 23121
This theorem is referenced by: (None)
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