Theorem gneispacess2 45240

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 45239	. . . 4 ⊢ (𝐹 ∈ 𝐴 → ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))
3		fveq2 6886	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝐹‘𝑝) = (𝐹‘𝑃))
4	3	eleq2d 2846	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑠 ∈ (𝐹‘𝑝) ↔ 𝑠 ∈ (𝐹‘𝑃)))
5	4	imbi2d 343	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ((𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)) ↔ (𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃))))
6	5	ralbidv 3185	. . . . . 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 3334	. . . . 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 3573	. . . 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 18	. . 3 ⊢ (𝐹 ∈ 𝐴 → (𝑃 ∈ dom 𝐹 → ∀𝑛 ∈ (𝐹‘𝑃)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃))))
10		sseq1 3933	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛 ⊆ 𝑠 ↔ 𝑁 ⊆ 𝑠))
11	10	imbi1d 344	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃)) ↔ (𝑁 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃))))
12	11	ralbidv 3185	. . . . . 6 ⊢ (𝑛 = 𝑁 → (∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃)) ↔ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑁 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃))))
13	12	rspccv 3573	. . . . 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 3934	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑁 ⊆ 𝑠 ↔ 𝑁 ⊆ 𝑆))
15		eleq1 2848	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ (𝐹‘𝑃) ↔ 𝑆 ∈ (𝐹‘𝑃)))
16	14, 15	imbi12d 347	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑁 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃)) ↔ (𝑁 ⊆ 𝑆 → 𝑆 ∈ (𝐹‘𝑃))))
17	16	rspccv 3573	. . . . 5 ⊢ (∀𝑠 ∈ 𝒫 dom 𝐹(𝑁 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃)) → (𝑆 ∈ 𝒫 dom 𝐹 → (𝑁 ⊆ 𝑆 → 𝑆 ∈ (𝐹‘𝑃))))
18	13, 17	syl6 36	. . . 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 1367	. . 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 36	. 2 ⊢ (𝐹 ∈ 𝐴 → (𝑃 ∈ dom 𝐹 → ((𝑁 ∈ (𝐹‘𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹 ∧ 𝑁 ⊆ 𝑆) → 𝑆 ∈ (𝐹‘𝑃))))
21	20	imp31 423	1 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹) ∧ (𝑁 ∈ (𝐹‘𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹 ∧ 𝑁 ⊆ 𝑆)) → 𝑆 ∈ (𝐹‘𝑃))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076   ⊆ wss 3895   ∖ cdif 4045  ∅c0 4278  𝒫 cpw 4556  {csn 4584  dom cdm 5650  ⟶wf 6535  ‘cfv 6539

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547

This theorem is used by: (None)