Theorem gneispacess2 41756

Description: All supersets of a neighborhood of a point (limited to the domain of the neighborhood space) are also neighborhoods of 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
gneispacess2	⊢ (((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹) ∧ (𝑁 ∈ (𝐹‘𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹 ∧ 𝑁 ⊆ 𝑆)) → 𝑆 ∈ (𝐹‘𝑃))

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

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

Proof of Theorem gneispacess2

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	gneispacess 41755	. . . 4 ⊢ (𝐹 ∈ 𝐴 → ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))
3		fveq2 6774	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝐹‘𝑝) = (𝐹‘𝑃))
4	3	eleq2d 2824	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑠 ∈ (𝐹‘𝑝) ↔ 𝑠 ∈ (𝐹‘𝑃)))
5	4	imbi2d 341	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ((𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)) ↔ (𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃))))
6	5	ralbidv 3112	. . . . . 6 ⊢ (𝑝 = 𝑃 → (∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)) ↔ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃))))
7	3, 6	raleqbidv 3336	. . . . 5 ⊢ (𝑝 = 𝑃 → (∀𝑛 ∈ (𝐹‘𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)) ↔ ∀𝑛 ∈ (𝐹‘𝑃)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃))))
8	7	rspccv 3558	. . . 4 ⊢ (∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)) → (𝑃 ∈ dom 𝐹 → ∀𝑛 ∈ (𝐹‘𝑃)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃))))
9	2, 8	syl 17	. . 3 ⊢ (𝐹 ∈ 𝐴 → (𝑃 ∈ dom 𝐹 → ∀𝑛 ∈ (𝐹‘𝑃)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃))))
10		sseq1 3946	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛 ⊆ 𝑠 ↔ 𝑁 ⊆ 𝑠))
11	10	imbi1d 342	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃)) ↔ (𝑁 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃))))
12	11	ralbidv 3112	. . . . . 6 ⊢ (𝑛 = 𝑁 → (∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃)) ↔ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑁 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃))))
13	12	rspccv 3558	. . . . 5 ⊢ (∀𝑛 ∈ (𝐹‘𝑃)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃)) → (𝑁 ∈ (𝐹‘𝑃) → ∀𝑠 ∈ 𝒫 dom 𝐹(𝑁 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃))))
14		sseq2 3947	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑁 ⊆ 𝑠 ↔ 𝑁 ⊆ 𝑆))
15		eleq1 2826	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ (𝐹‘𝑃) ↔ 𝑆 ∈ (𝐹‘𝑃)))
16	14, 15	imbi12d 345	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑁 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃)) ↔ (𝑁 ⊆ 𝑆 → 𝑆 ∈ (𝐹‘𝑃))))
17	16	rspccv 3558	. . . . 5 ⊢ (∀𝑠 ∈ 𝒫 dom 𝐹(𝑁 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃)) → (𝑆 ∈ 𝒫 dom 𝐹 → (𝑁 ⊆ 𝑆 → 𝑆 ∈ (𝐹‘𝑃))))
18	13, 17	syl6 35	. . . 4 ⊢ (∀𝑛 ∈ (𝐹‘𝑃)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃)) → (𝑁 ∈ (𝐹‘𝑃) → (𝑆 ∈ 𝒫 dom 𝐹 → (𝑁 ⊆ 𝑆 → 𝑆 ∈ (𝐹‘𝑃)))))
19	18	3impd 1347	. . 3 ⊢ (∀𝑛 ∈ (𝐹‘𝑃)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃)) → ((𝑁 ∈ (𝐹‘𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹 ∧ 𝑁 ⊆ 𝑆) → 𝑆 ∈ (𝐹‘𝑃)))
20	9, 19	syl6 35	. 2 ⊢ (𝐹 ∈ 𝐴 → (𝑃 ∈ dom 𝐹 → ((𝑁 ∈ (𝐹‘𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹 ∧ 𝑁 ⊆ 𝑆) → 𝑆 ∈ (𝐹‘𝑃))))
21	20	imp31 418	1 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹) ∧ (𝑁 ∈ (𝐹‘𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹 ∧ 𝑁 ⊆ 𝑆)) → 𝑆 ∈ (𝐹‘𝑃))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064   ∖ cdif 3884   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561  dom cdm 5589  ⟶wf 6429  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441

This theorem is referenced by: (None)