Theorem gneispaceel2 43639

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 43638	. . . 4 ⊢ (𝐹 ∈ 𝐴 → ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)𝑝 ∈ 𝑛)
3		fveq2 6892	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝐹‘𝑝) = (𝐹‘𝑃))
4		eleq1 2813	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∈ 𝑛 ↔ 𝑃 ∈ 𝑛))
5	3, 4	raleqbidv 3330	. . . . 5 ⊢ (𝑝 = 𝑃 → (∀𝑛 ∈ (𝐹‘𝑝)𝑝 ∈ 𝑛 ↔ ∀𝑛 ∈ (𝐹‘𝑃)𝑃 ∈ 𝑛))
6	5	rspccv 3598	. . . 4 ⊢ (∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)𝑝 ∈ 𝑛 → (𝑃 ∈ dom 𝐹 → ∀𝑛 ∈ (𝐹‘𝑃)𝑃 ∈ 𝑛))
7	2, 6	syl 17	. . 3 ⊢ (𝐹 ∈ 𝐴 → (𝑃 ∈ dom 𝐹 → ∀𝑛 ∈ (𝐹‘𝑃)𝑃 ∈ 𝑛))
8		eleq2 2814	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑃 ∈ 𝑛 ↔ 𝑃 ∈ 𝑁))
9	8	rspccv 3598	. . 3 ⊢ (∀𝑛 ∈ (𝐹‘𝑃)𝑃 ∈ 𝑛 → (𝑁 ∈ (𝐹‘𝑃) → 𝑃 ∈ 𝑁))
10	7, 9	syl6 35	. 2 ⊢ (𝐹 ∈ 𝐴 → (𝑃 ∈ dom 𝐹 → (𝑁 ∈ (𝐹‘𝑃) → 𝑃 ∈ 𝑁)))
11	10	3imp 1108	1 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹 ∧ 𝑁 ∈ (𝐹‘𝑃)) → 𝑃 ∈ 𝑁)

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2702  ∀wral 3051   ∖ cdif 3936   ⊆ wss 3939  ∅c0 4318  𝒫 cpw 4598  {csn 4624  dom cdm 5672  ⟶wf 6539  ‘cfv 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551

This theorem is referenced by: (None)