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Theorem gneispaceel2 42508
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 42507 . . . 4 (𝐹𝐴 → ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)𝑝𝑛)
3 fveq2 6846 . . . . . 6 (𝑝 = 𝑃 → (𝐹𝑝) = (𝐹𝑃))
4 eleq1 2822 . . . . . 6 (𝑝 = 𝑃 → (𝑝𝑛𝑃𝑛))
53, 4raleqbidv 3318 . . . . 5 (𝑝 = 𝑃 → (∀𝑛 ∈ (𝐹𝑝)𝑝𝑛 ↔ ∀𝑛 ∈ (𝐹𝑃)𝑃𝑛))
65rspccv 3580 . . . 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 2823 . . . 4 (𝑛 = 𝑁 → (𝑃𝑛𝑃𝑁))
98rspccv 3580 . . 3 (∀𝑛 ∈ (𝐹𝑃)𝑃𝑛 → (𝑁 ∈ (𝐹𝑃) → 𝑃𝑁))
107, 9syl6 35 . 2 (𝐹𝐴 → (𝑃 ∈ dom 𝐹 → (𝑁 ∈ (𝐹𝑃) → 𝑃𝑁)))
11103imp 1112 1 ((𝐹𝐴𝑃 ∈ dom 𝐹𝑁 ∈ (𝐹𝑃)) → 𝑃𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wral 3061  cdif 3911  wss 3914  c0 4286  𝒫 cpw 4564  {csn 4590  dom cdm 5637  wf 6496  cfv 6500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508
This theorem is referenced by: (None)
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