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Theorem neissex 21419
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 21400 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑥𝐽 (𝑆𝑥𝑥𝑁))
2 opnneiss 21410 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑆𝑥) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
323expb 1113 . . . 4 ((𝐽 ∈ Top ∧ (𝑥𝐽𝑆𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
43adantrrr 721 . . 3 ((𝐽 ∈ Top ∧ (𝑥𝐽 ∧ (𝑆𝑥𝑥𝑁))) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
54adantlr 711 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽 ∧ (𝑆𝑥𝑥𝑁))) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
6 simplll 771 . . . . . 6 ((((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽𝑥𝑁)) ∧ 𝑦𝑥) → 𝐽 ∈ Top)
7 simpll 763 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) → 𝐽 ∈ Top)
8 simpr 485 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) → 𝑥𝐽)
9 eqid 2795 . . . . . . . . . . . 12 𝐽 = 𝐽
109neii1 21398 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 𝐽)
1110adantr 481 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝑥𝐽) → 𝑁 𝐽)
129opnssneib 21407 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥𝐽𝑁 𝐽) → (𝑥𝑁𝑁 ∈ ((nei‘𝐽)‘𝑥)))
137, 8, 11, 12syl3anc 1364 . . . . . . . . 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 21401 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑥) ∧ 𝑦𝑥) → 𝑁 ∈ ((nei‘𝐽)‘𝑦))
196, 16, 17, 18syl3anc 1364 . . . . 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 720 . . 3 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽 ∧ (𝑆𝑥𝑥𝑁))) → (𝑦𝑥𝑁 ∈ ((nei‘𝐽)‘𝑦)))
2221alrimiv 1905 . 2 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) ∧ (𝑥𝐽 ∧ (𝑆𝑥𝑥𝑁))) → ∀𝑦(𝑦𝑥𝑁 ∈ ((nei‘𝐽)‘𝑦)))
231, 5, 22reximssdv 3239 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 207  wa 396  wal 1520  wcel 2081  wrex 3106  wss 3859   cuni 4745  cfv 6225  Topctop 21185  neicnei 21389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-top 21186  df-nei 21390
This theorem is referenced by: (None)
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