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Theorem gneispaceel2 43476
Description: Every neighborhood of a point in a generic neighborhood space contains that point. (Contributed by RP, 15-Apr-2021.)
Hypothesis
Ref Expression
gneispace.a 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}
Assertion
Ref Expression
gneispaceel2 ((𝐹𝐴𝑃 ∈ dom 𝐹𝑁 ∈ (𝐹𝑃)) → 𝑃𝑁)
Distinct variable groups:   𝑛,𝐹,𝑝,𝑓   𝐹,𝑠,𝑓   𝑓,𝑛,𝑝   𝑃,𝑝,𝑛   𝑛,𝑁
Allowed substitution hints:   𝐴(𝑓,𝑛,𝑠,𝑝)   𝑃(𝑓,𝑠)   𝑁(𝑓,𝑠,𝑝)

Proof of Theorem gneispaceel2
StepHypRef Expression
1 gneispace.a . . . . 5 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}
21gneispaceel 43475 . . . 4 (𝐹𝐴 → ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)𝑝𝑛)
3 fveq2 6885 . . . . . 6 (𝑝 = 𝑃 → (𝐹𝑝) = (𝐹𝑃))
4 eleq1 2815 . . . . . 6 (𝑝 = 𝑃 → (𝑝𝑛𝑃𝑛))
53, 4raleqbidv 3336 . . . . 5 (𝑝 = 𝑃 → (∀𝑛 ∈ (𝐹𝑝)𝑝𝑛 ↔ ∀𝑛 ∈ (𝐹𝑃)𝑃𝑛))
65rspccv 3603 . . . 4 (∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)𝑝𝑛 → (𝑃 ∈ dom 𝐹 → ∀𝑛 ∈ (𝐹𝑃)𝑃𝑛))
72, 6syl 17 . . 3 (𝐹𝐴 → (𝑃 ∈ dom 𝐹 → ∀𝑛 ∈ (𝐹𝑃)𝑃𝑛))
8 eleq2 2816 . . . 4 (𝑛 = 𝑁 → (𝑃𝑛𝑃𝑁))
98rspccv 3603 . . 3 (∀𝑛 ∈ (𝐹𝑃)𝑃𝑛 → (𝑁 ∈ (𝐹𝑃) → 𝑃𝑁))
107, 9syl6 35 . 2 (𝐹𝐴 → (𝑃 ∈ dom 𝐹 → (𝑁 ∈ (𝐹𝑃) → 𝑃𝑁)))
11103imp 1108 1 ((𝐹𝐴𝑃 ∈ dom 𝐹𝑁 ∈ (𝐹𝑃)) → 𝑃𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  {cab 2703  wral 3055  cdif 3940  wss 3943  c0 4317  𝒫 cpw 4597  {csn 4623  dom cdm 5669  wf 6533  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545
This theorem is referenced by: (None)
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