Theorem gneispace0nelrn2 44098

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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}

Assertion

Ref	Expression
gneispace0nelrn2	⊢ ((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹) → (𝐹‘𝑃) ≠ ∅)

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

Step	Hyp	Ref	Expression
1		gneispace.a	. . . 4 ⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}
2	1	gneispace0nelrn 44097	. . 3 ⊢ (𝐹 ∈ 𝐴 → ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅)
3		fveq2 6915	. . . . 5 ⊢ (𝑝 = 𝑃 → (𝐹‘𝑝) = (𝐹‘𝑃))
4	3	neeq1d 3006	. . . 4 ⊢ (𝑝 = 𝑃 → ((𝐹‘𝑝) ≠ ∅ ↔ (𝐹‘𝑃) ≠ ∅))
5	4	rspccv 3632	. . 3 ⊢ (∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅ → (𝑃 ∈ dom 𝐹 → (𝐹‘𝑃) ≠ ∅))
6	2, 5	syl 17	. 2 ⊢ (𝐹 ∈ 𝐴 → (𝑃 ∈ dom 𝐹 → (𝐹‘𝑃) ≠ ∅))
7	6	imp 406	1 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹) → (𝐹‘𝑃) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717   ≠ wne 2946  ∀wral 3067   ∖ cdif 3973   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  {csn 4648  dom cdm 5695  ⟶wf 6564  ‘cfv 6568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-iota 6520  df-fun 6570  df-fn 6571  df-f 6572  df-fv 6576

This theorem is referenced by: (None)