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Theorem llynlly 21474
Description: A locally 𝐴 space is n-locally 𝐴: the "n-locally" predicate is the weaker notion. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llynlly (𝐽 ∈ Locally 𝐴𝐽 ∈ 𝑛-Locally 𝐴)

Proof of Theorem llynlly
Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 21469 . 2 (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)
2 llyi 21471 . . . . 5 ((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) → ∃𝑢𝐽 (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
3 simpl1 1225 . . . . . . . . . . 11 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Locally 𝐴)
43, 1syl 17 . . . . . . . . . 10 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
5 simprl 811 . . . . . . . . . 10 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢𝐽)
6 simprr2 1272 . . . . . . . . . 10 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑦𝑢)
7 opnneip 21117 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑢𝐽𝑦𝑢) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
84, 5, 6, 7syl3anc 1473 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
9 simprr1 1270 . . . . . . . . . 10 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢𝑥)
10 selpw 4301 . . . . . . . . . 10 (𝑢 ∈ 𝒫 𝑥𝑢𝑥)
119, 10sylibr 224 . . . . . . . . 9 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢 ∈ 𝒫 𝑥)
128, 11elind 3933 . . . . . . . 8 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥))
13 simprr3 1274 . . . . . . . 8 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝐽t 𝑢) ∈ 𝐴)
1412, 13jca 555 . . . . . . 7 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝐽t 𝑢) ∈ 𝐴))
1514ex 449 . . . . . 6 ((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) → ((𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴)) → (𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥) ∧ (𝐽t 𝑢) ∈ 𝐴)))
1615reximdv2 3144 . . . . 5 ((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) → (∃𝑢𝐽 (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
172, 16mpd 15 . . . 4 ((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
18173expb 1113 . . 3 ((𝐽 ∈ Locally 𝐴 ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
1918ralrimivva 3101 . 2 (𝐽 ∈ Locally 𝐴 → ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
20 isnlly 21466 . 2 (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
211, 19, 20sylanbrc 701 1 (𝐽 ∈ Locally 𝐴𝐽 ∈ 𝑛-Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072  wcel 2131  wral 3042  wrex 3043  cin 3706  wss 3707  𝒫 cpw 4294  {csn 4313  cfv 6041  (class class class)co 6805  t crest 16275  Topctop 20892  neicnei 21095  Locally clly 21461  𝑛-Locally cnlly 21462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-top 20893  df-nei 21096  df-lly 21463  df-nlly 21464
This theorem is referenced by:  llyssnlly  21475  symgtgp  22098
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