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Theorem isnlly 23366
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 6891 . . . . . . 7 (𝑗 = 𝐽 → (nei‘𝑗) = (nei‘𝐽))
21fveq1d 6893 . . . . . 6 (𝑗 = 𝐽 → ((nei‘𝑗)‘{𝑦}) = ((nei‘𝐽)‘{𝑦}))
32ineq1d 4207 . . . . 5 (𝑗 = 𝐽 → (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥) = (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥))
4 oveq1 7421 . . . . . 6 (𝑗 = 𝐽 → (𝑗t 𝑢) = (𝐽t 𝑢))
54eleq1d 2814 . . . . 5 (𝑗 = 𝐽 → ((𝑗t 𝑢) ∈ 𝐴 ↔ (𝐽t 𝑢) ∈ 𝐴))
63, 5rexeqbidv 3339 . . . 4 (𝑗 = 𝐽 → (∃𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴 ↔ ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
76ralbidv 3173 . . 3 (𝑗 = 𝐽 → (∀𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴 ↔ ∀𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
87raleqbi1dv 3329 . 2 (𝑗 = 𝐽 → (∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴 ↔ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
9 df-nlly 23364 . 2 𝑛-Locally 𝐴 = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑢 ∈ (((nei‘𝑗)‘{𝑦}) ∩ 𝒫 𝑥)(𝑗t 𝑢) ∈ 𝐴}
108, 9elrab2 3684 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 205  wa 395   = wceq 1534  wcel 2099  wral 3057  wrex 3066  cin 3944  𝒫 cpw 4598  {csn 4624  cfv 6542  (class class class)co 7414  t crest 17395  Topctop 22788  neicnei 22994  𝑛-Locally cnlly 23362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417  df-nlly 23364
This theorem is referenced by:  nllytop  23370  nllyi  23372  llynlly  23374  nllyss  23377  nllyrest  23383  nllyidm  23386  hausllycmp  23391  cldllycmp  23392  txnlly  23534  cnllycmp  24875
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