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Theorem gneispace0nelrn2 44089
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 44088 . . 3 (𝐹𝐴 → ∀𝑝 ∈ dom 𝐹(𝐹𝑝) ≠ ∅)
3 fveq2 6901 . . . . 5 (𝑝 = 𝑃 → (𝐹𝑝) = (𝐹𝑃))
43neeq1d 2996 . . . 4 (𝑝 = 𝑃 → ((𝐹𝑝) ≠ ∅ ↔ (𝐹𝑃) ≠ ∅))
54rspccv 3619 . . 3 (∀𝑝 ∈ dom 𝐹(𝐹𝑝) ≠ ∅ → (𝑃 ∈ dom 𝐹 → (𝐹𝑃) ≠ ∅))
62, 5syl 17 . 2 (𝐹𝐴 → (𝑃 ∈ dom 𝐹 → (𝐹𝑃) ≠ ∅))
76imp 406 1 ((𝐹𝐴𝑃 ∈ dom 𝐹) → (𝐹𝑃) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1535  wcel 2104  {cab 2710  wne 2936  wral 3057  cdif 3960  wss 3963  c0 4339  𝒫 cpw 4604  {csn 4630  dom cdm 5683  wf 6554  cfv 6558
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 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-fv 6566
This theorem is referenced by: (None)
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