Theorem gneispace0nelrn2 44382

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 44381	. . 3 ⊢ (𝐹 ∈ 𝐴 → ∀𝑝 ∈ dom 𝐹(𝐹‘𝑝) ≠ ∅)
3		fveq2 6834	. . . . 5 ⊢ (𝑝 = 𝑃 → (𝐹‘𝑝) = (𝐹‘𝑃))
4	3	neeq1d 2991	. . . 4 ⊢ (𝑝 = 𝑃 → ((𝐹‘𝑝) ≠ ∅ ↔ (𝐹‘𝑃) ≠ ∅))
5	4	rspccv 3573	. . 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 1541   ∈ wcel 2113  {cab 2714   ≠ wne 2932  ∀wral 3051   ∖ cdif 3898   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  {csn 4580  dom cdm 5624  ⟶wf 6488  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500

This theorem is referenced by: (None)