Theorem gneispaceel2 44992

Description: Every neighborhood of a point in a generic neighborhood space contains that point. (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
gneispaceel2	⊢ ((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹 ∧ 𝑁 ∈ (𝐹‘𝑃)) → 𝑃 ∈ 𝑁)

Distinct variable groups:   𝑛,𝐹,𝑝,𝑓   𝐹,𝑠,𝑓   𝑓,𝑛,𝑝   𝑃,𝑝,𝑛   𝑛,𝑁

Allowed substitution hints:   𝐴(𝑓, 𝑛, 𝑠, 𝑝)   𝑃(𝑓, 𝑠)   𝑁(𝑓, 𝑠, 𝑝)

Proof of Theorem gneispaceel2

Step	Hyp	Ref	Expression
1		gneispace.a	. . . . 5 ⊢ 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	gneispaceel 44991	. . . 4 ⊢ (𝐹 ∈ 𝐴 → ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)𝑝 ∈ 𝑛)
3		fveq2 6882	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝐹‘𝑝) = (𝐹‘𝑃))
4		eleq1 2850	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∈ 𝑛 ↔ 𝑃 ∈ 𝑛))
5	3, 4	raleqbidv 3336	. . . . 5 ⊢ (𝑝 = 𝑃 → (∀𝑛 ∈ (𝐹‘𝑝)𝑝 ∈ 𝑛 ↔ ∀𝑛 ∈ (𝐹‘𝑃)𝑃 ∈ 𝑛))
6	5	rspccv 3576	. . . 4 ⊢ (∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)𝑝 ∈ 𝑛 → (𝑃 ∈ dom 𝐹 → ∀𝑛 ∈ (𝐹‘𝑃)𝑃 ∈ 𝑛))
7	2, 6	syl 18	. . 3 ⊢ (𝐹 ∈ 𝐴 → (𝑃 ∈ dom 𝐹 → ∀𝑛 ∈ (𝐹‘𝑃)𝑃 ∈ 𝑛))
8		eleq2 2851	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑃 ∈ 𝑛 ↔ 𝑃 ∈ 𝑁))
9	8	rspccv 3576	. . 3 ⊢ (∀𝑛 ∈ (𝐹‘𝑃)𝑃 ∈ 𝑛 → (𝑁 ∈ (𝐹‘𝑃) → 𝑃 ∈ 𝑁))
10	7, 9	syl6 36	. 2 ⊢ (𝐹 ∈ 𝐴 → (𝑃 ∈ dom 𝐹 → (𝑁 ∈ (𝐹‘𝑃) → 𝑃 ∈ 𝑁)))
11	10	3imp 1128	1 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹 ∧ 𝑁 ∈ (𝐹‘𝑃)) → 𝑃 ∈ 𝑁)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2740  ∀wral 3078   ∖ cdif 3899   ⊆ wss 3902  ∅c0 4282  𝒫 cpw 4560  {csn 4587  dom cdm 5659  ⟶wf 6533  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545

This theorem is used by: (None)