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Theorem isnlly 21494
Description: The property of being an n-locally 𝐴 topological space. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
isnlly (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
Distinct variable groups:   𝑥,𝑢,𝑦,𝐴   𝑢,𝐽,𝑥,𝑦

Proof of Theorem isnlly
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6353 . . . . . . 7 (𝑗 = 𝐽 → (nei‘𝑗) = (nei‘𝐽))
21fveq1d 6355 . . . . . 6 (𝑗 = 𝐽 → ((nei‘𝑗)‘{𝑦}) = ((nei‘𝐽)‘{𝑦}))
32ineq1d 3956 . . . . 5 (𝑗 = 𝐽 → (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) = (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥))
4 oveq1 6821 . . . . . 6 (𝑗 = 𝐽 → (𝑗t 𝑢) = (𝐽t 𝑢))
54eleq1d 2824 . . . . 5 (𝑗 = 𝐽 → ((𝑗t 𝑢) ∈ 𝐴 ↔ (𝐽t 𝑢) ∈ 𝐴))
63, 5rexeqbidv 3292 . . . 4 (𝑗 = 𝐽 → (∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴 ↔ ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
76ralbidv 3124 . . 3 (𝑗 = 𝐽 → (∀𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴 ↔ ∀𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
87raleqbi1dv 3285 . 2 (𝑗 = 𝐽 → (∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴 ↔ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
9 df-nlly 21492 . 2 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴}
108, 9elrab2 3507 1 (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  cin 3714  𝒫 cpw 4302  {csn 4321  cfv 6049  (class class class)co 6814  t crest 16303  Topctop 20920  neicnei 21123  𝑛-Locally cnlly 21490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6817  df-nlly 21492
This theorem is referenced by:  nllytop  21498  nllyi  21500  llynlly  21502  nllyss  21505  nllyrest  21511  nllyidm  21514  hausllycmp  21519  cldllycmp  21520  txnlly  21662  cnllycmp  22976
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