Theorem gneispacess2 44686

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 44685	. . . 4 ⊢ (𝐹 ∈ 𝐴 → ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))
3		fveq2 6863	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝐹‘𝑝) = (𝐹‘𝑃))
4	3	eleq2d 2847	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑠 ∈ (𝐹‘𝑝) ↔ 𝑠 ∈ (𝐹‘𝑃)))
5	4	imbi2d 342	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ((𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)) ↔ (𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃))))
6	5	ralbidv 3184	. . . . . 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 3335	. . . . 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 3578	. . . 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 3961	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛 ⊆ 𝑠 ↔ 𝑁 ⊆ 𝑠))
11	10	imbi1d 343	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃)) ↔ (𝑁 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃))))
12	11	ralbidv 3184	. . . . . 6 ⊢ (𝑛 = 𝑁 → (∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃)) ↔ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑁 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃))))
13	12	rspccv 3578	. . . . 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 3962	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑁 ⊆ 𝑠 ↔ 𝑁 ⊆ 𝑆))
15		eleq1 2849	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ (𝐹‘𝑃) ↔ 𝑆 ∈ (𝐹‘𝑃)))
16	14, 15	imbi12d 346	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝑁 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑃)) ↔ (𝑁 ⊆ 𝑆 → 𝑆 ∈ (𝐹‘𝑃))))
17	16	rspccv 3578	. . . . 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 1361	. . 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 421	1 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝑃 ∈ dom 𝐹) ∧ (𝑁 ∈ (𝐹‘𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹 ∧ 𝑁 ⊆ 𝑆)) → 𝑆 ∈ (𝐹‘𝑃))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  {cab 2739  ∀wral 3075   ∖ cdif 3901   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4554  {csn 4581  dom cdm 5645  ⟶wf 6513  ‘cfv 6517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fv 6525

This theorem is referenced by: (None)