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Theorem gneispace0nelrn2 44132
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 44131 . . 3 (𝐹𝐴 → ∀𝑝 ∈ dom 𝐹(𝐹𝑝) ≠ ∅)
3 fveq2 6904 . . . . 5 (𝑝 = 𝑃 → (𝐹𝑝) = (𝐹𝑃))
43neeq1d 2999 . . . 4 (𝑝 = 𝑃 → ((𝐹𝑝) ≠ ∅ ↔ (𝐹𝑃) ≠ ∅))
54rspccv 3618 . . 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 1540  wcel 2108  {cab 2713  wne 2939  wral 3060  cdif 3947  wss 3950  c0 4332  𝒫 cpw 4598  {csn 4624  dom cdm 5683  wf 6555  cfv 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-fv 6567
This theorem is referenced by: (None)
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