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Theorem neissex 21139
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 21120 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥𝐽 (𝑆𝑥𝑥𝑁))
2 opnneiss 21130 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑆𝑥) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
323expb 1142 . . . 4 ((𝐽 ∈ Top ∧ (𝑥𝐽𝑆𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
43adantrrr 707 . . 3 ((𝐽 ∈ Top ∧ (𝑥𝐽 ∧ (𝑆𝑥𝑥𝑁))) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
54adantlr 697 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽 ∧ (𝑆𝑥𝑥𝑁))) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
6 simplll 782 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) ∧ 𝑦𝑥) → 𝐽 ∈ Top)
7 simpll 774 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) → 𝐽 ∈ Top)
8 simpr 473 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) → 𝑥𝐽)
9 eqid 2802 . . . . . . . . . . . 12 𝐽 = 𝐽
109neii1 21118 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 𝐽)
1110adantr 468 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) → 𝑁 𝐽)
129opnssneib 21127 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑁 𝐽) → (𝑥𝑁𝑁 ∈ ((nei‘𝐽)‘𝑥)))
137, 8, 11, 12syl3anc 1483 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) → (𝑥𝑁𝑁 ∈ ((nei‘𝐽)‘𝑥)))
1413biimpa 464 . . . . . . . 8 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) ∧ 𝑥𝑁) → 𝑁 ∈ ((nei‘𝐽)‘𝑥))
1514anasss 454 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) → 𝑁 ∈ ((nei‘𝐽)‘𝑥))
1615adantr 468 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) ∧ 𝑦𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑥))
17 simpr 473 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) ∧ 𝑦𝑥) → 𝑦𝑥)
18 neiss 21121 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑥) ∧ 𝑦𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑦))
196, 16, 17, 18syl3anc 1483 . . . . 5 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) ∧ 𝑦𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑦))
2019ex 399 . . . 4 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) → (𝑦𝑥𝑁 ∈ ((nei‘𝐽)‘𝑦)))
2120adantrrl 706 . . 3 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽 ∧ (𝑆𝑥𝑥𝑁))) → (𝑦𝑥𝑁 ∈ ((nei‘𝐽)‘𝑦)))
2221alrimiv 2017 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽 ∧ (𝑆𝑥𝑥𝑁))) → ∀𝑦(𝑦𝑥𝑁 ∈ ((nei‘𝐽)‘𝑦)))
231, 5, 22reximssdv 3202 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 197  wa 384  wal 1635  wcel 2155  wrex 3093  wss 3763   cuni 4623  cfv 6095  Topctop 20905  neicnei 21109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-id 5213  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-top 20906  df-nei 21110
This theorem is referenced by: (None)
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