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Theorem gneispace0nelrn2 41732
Description: A generic neighborhood space has a nonempty set of neighborhoods for every point in its domain. (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
gneispace0nelrn2 ((𝐹𝐴𝑃 ∈ dom 𝐹) → (𝐹𝑃) ≠ ∅)
Distinct variable groups:   𝑛,𝐹,𝑝,𝑓   𝐹,𝑠,𝑓   𝑓,𝑛,𝑝   𝑃,𝑝,𝑛
Allowed substitution hints:   𝐴(𝑓,𝑛,𝑠,𝑝)   𝑃(𝑓,𝑠)

Proof of Theorem gneispace0nelrn2
StepHypRef Expression
1 gneispace.a . . . 4 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}
21gneispace0nelrn 41731 . . 3 (𝐹𝐴 → ∀𝑝 ∈ dom 𝐹(𝐹𝑝) ≠ ∅)
3 fveq2 6766 . . . . 5 (𝑝 = 𝑃 → (𝐹𝑝) = (𝐹𝑃))
43neeq1d 3003 . . . 4 (𝑝 = 𝑃 → ((𝐹𝑝) ≠ ∅ ↔ (𝐹𝑃) ≠ ∅))
54rspccv 3556 . . 3 (∀𝑝 ∈ dom 𝐹(𝐹𝑝) ≠ ∅ → (𝑃 ∈ dom 𝐹 → (𝐹𝑃) ≠ ∅))
62, 5syl 17 . 2 (𝐹𝐴 → (𝑃 ∈ dom 𝐹 → (𝐹𝑃) ≠ ∅))
76imp 407 1 ((𝐹𝐴𝑃 ∈ dom 𝐹) → (𝐹𝑃) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  {cab 2715  wne 2943  wral 3064  cdif 3883  wss 3886  c0 4256  𝒫 cpw 4533  {csn 4561  dom cdm 5584  wf 6422  cfv 6426
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-fv 6434
This theorem is referenced by: (None)
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