Theorem neiptopreu 22192

Description: If, to each element 𝑃 of a set 𝑋, we associate a set (𝑁‘𝑃) fulfilling Properties V_i, V_ii, V_iii and Property V_iv of [BourbakiTop1] p. I.2. , corresponding to ssnei 22169, innei 22184, elnei 22170 and neissex 22186, then there is a unique topology 𝑗 such that for any point 𝑝, (𝑁‘𝑝) is the set of neighborhoods of 𝑝. Proposition 2 of [BourbakiTop1] p. I.3. This can be used to build a topology from a set of neighborhoods. Note that innei 22184 uses binary intersections whereas Property V_ii mentions finite intersections (which includes the empty intersection of subsets of 𝑋, which is equal to 𝑋), so we add the hypothesis that 𝑋 is a neighborhood of all points. TODO: when df-fi 9100 includes the empty intersection, remove that extra hypothesis. (Contributed by Thierry Arnoux, 6-Jan-2018.)

Hypotheses

Ref	Expression
neiptop.o	⊢ 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}
neiptop.0	⊢ (𝜑 → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
neiptop.1	⊢ ((((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))
neiptop.2	⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))
neiptop.3	⊢ (((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)
neiptop.4	⊢ (((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑏 ∈ (𝑁‘𝑝)∀𝑞 ∈ 𝑏 𝑎 ∈ (𝑁‘𝑞))
neiptop.5	⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))

Assertion

Ref	Expression
neiptopreu	⊢ (𝜑 → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})))

Distinct variable groups:   𝑝,𝑎,𝑁   𝑋,𝑎,𝑏,𝑝   𝐽,𝑎,𝑝   𝑋,𝑝   𝜑,𝑝   𝑁,𝑏   𝑋,𝑏   𝜑,𝑎,𝑏,𝑞,𝑝   𝑁,𝑝,𝑞   𝑋,𝑞   𝜑,𝑞   𝑗,𝑎,𝑏,𝐽,𝑝   𝑗,𝑞,𝑁   𝑗,𝑋   𝜑,𝑗

Allowed substitution hint:   𝐽(𝑞)

Proof of Theorem neiptopreu

Step	Hyp	Ref	Expression
1		neiptop.o	. . . . 5 ⊢ 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}
2		neiptop.0	. . . . 5 ⊢ (𝜑 → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
3		neiptop.1	. . . . 5 ⊢ ((((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))
4		neiptop.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))
5		neiptop.3	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)
6		neiptop.4	. . . . 5 ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑏 ∈ (𝑁‘𝑝)∀𝑞 ∈ 𝑏 𝑎 ∈ (𝑁‘𝑞))
7		neiptop.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))
8	1, 2, 3, 4, 5, 6, 7	neiptoptop 22190	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Top)
9		toptopon2 21975	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
10	8, 9	sylib 217	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽))
11	1, 2, 3, 4, 5, 6, 7	neiptopuni 22189	. . . 4 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
12	11	fveq2d 6760	. . 3 ⊢ (𝜑 → (TopOn‘𝑋) = (TopOn‘∪ 𝐽))
13	10, 12	eleqtrrd 2842	. 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
14	1, 2, 3, 4, 5, 6, 7	neiptopnei 22191	. 2 ⊢ (𝜑 → 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝐽)‘{𝑝})))
15		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑝(𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋))
16		nfmpt1 5178	. . . . . . . . . . 11 ⊢ Ⅎ𝑝(𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))
17	16	nfeq2 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑝 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))
18	15, 17	nfan 1903	. . . . . . . . 9 ⊢ Ⅎ𝑝((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
19		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑝 𝑏 ⊆ 𝑋
20	18, 19	nfan 1903	. . . . . . . 8 ⊢ Ⅎ𝑝(((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ⊆ 𝑋)
21		simpllr 772	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ⊆ 𝑋) ∧ 𝑝 ∈ 𝑏) → 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
22		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ⊆ 𝑋) → 𝑏 ⊆ 𝑋)
23	22	sselda 3917	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ⊆ 𝑋) ∧ 𝑝 ∈ 𝑏) → 𝑝 ∈ 𝑋)
24		id 22	. . . . . . . . . . . 12 ⊢ (𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
25		fvexd 6771	. . . . . . . . . . . 12 ⊢ ((𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ∧ 𝑝 ∈ 𝑋) → ((nei‘𝑗)‘{𝑝}) ∈ V)
26	24, 25	fvmpt2d 6870	. . . . . . . . . . 11 ⊢ ((𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ∧ 𝑝 ∈ 𝑋) → (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝}))
27	21, 23, 26	syl2anc 583	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ⊆ 𝑋) ∧ 𝑝 ∈ 𝑏) → (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝}))
28	27	eqcomd 2744	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ⊆ 𝑋) ∧ 𝑝 ∈ 𝑏) → ((nei‘𝑗)‘{𝑝}) = (𝑁‘𝑝))
29	28	eleq2d 2824	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ⊆ 𝑋) ∧ 𝑝 ∈ 𝑏) → (𝑏 ∈ ((nei‘𝑗)‘{𝑝}) ↔ 𝑏 ∈ (𝑁‘𝑝)))
30	20, 29	ralbida 3156	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ⊆ 𝑋) → (∀𝑝 ∈ 𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}) ↔ ∀𝑝 ∈ 𝑏 𝑏 ∈ (𝑁‘𝑝)))
31	30	pm5.32da 578	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → ((𝑏 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})) ↔ (𝑏 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑏 𝑏 ∈ (𝑁‘𝑝))))
32		toponss 21984	. . . . . . . . 9 ⊢ ((𝑗 ∈ (TopOn‘𝑋) ∧ 𝑏 ∈ 𝑗) → 𝑏 ⊆ 𝑋)
33	32	ad4ant24 750	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ∈ 𝑗) → 𝑏 ⊆ 𝑋)
34		topontop 21970	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (TopOn‘𝑋) → 𝑗 ∈ Top)
35	34	ad2antlr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → 𝑗 ∈ Top)
36		opnnei 22179	. . . . . . . . . 10 ⊢ (𝑗 ∈ Top → (𝑏 ∈ 𝑗 ↔ ∀𝑝 ∈ 𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})))
37	35, 36	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏 ∈ 𝑗 ↔ ∀𝑝 ∈ 𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})))
38	37	biimpa 476	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ∈ 𝑗) → ∀𝑝 ∈ 𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))
39	33, 38	jca 511	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ∈ 𝑗) → (𝑏 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})))
40	37	biimpar 477	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ ∀𝑝 ∈ 𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})) → 𝑏 ∈ 𝑗)
41	40	adantrl 712	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ (𝑏 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))) → 𝑏 ∈ 𝑗)
42	39, 41	impbida 797	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏 ∈ 𝑗 ↔ (𝑏 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))))
43	1	neipeltop 22188	. . . . . . 7 ⊢ (𝑏 ∈ 𝐽 ↔ (𝑏 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑏 𝑏 ∈ (𝑁‘𝑝)))
44	43	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏 ∈ 𝐽 ↔ (𝑏 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑏 𝑏 ∈ (𝑁‘𝑝))))
45	31, 42, 44	3bitr4d 310	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏 ∈ 𝑗 ↔ 𝑏 ∈ 𝐽))
46	45	eqrdv 2736	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → 𝑗 = 𝐽)
47	46	ex 412	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) → (𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑗 = 𝐽))
48	47	ralrimiva 3107	. 2 ⊢ (𝜑 → ∀𝑗 ∈ (TopOn‘𝑋)(𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑗 = 𝐽))
49		simpl 482	. . . . . . 7 ⊢ ((𝑗 = 𝐽 ∧ 𝑝 ∈ 𝑋) → 𝑗 = 𝐽)
50	49	fveq2d 6760	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑝 ∈ 𝑋) → (nei‘𝑗) = (nei‘𝐽))
51	50	fveq1d 6758	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑝 ∈ 𝑋) → ((nei‘𝑗)‘{𝑝}) = ((nei‘𝐽)‘{𝑝}))
52	51	mpteq2dva 5170	. . . 4 ⊢ (𝑗 = 𝐽 → (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) = (𝑝 ∈ 𝑋 ↦ ((nei‘𝐽)‘{𝑝})))
53	52	eqeq2d 2749	. . 3 ⊢ (𝑗 = 𝐽 → (𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝐽)‘{𝑝}))))
54	53	eqreu 3659	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝐽)‘{𝑝})) ∧ ∀𝑗 ∈ (TopOn‘𝑋)(𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑗 = 𝐽)) → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
55	13, 14, 48, 54	syl3anc 1369	1 ⊢ (𝜑 → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  ∃!wreu 3065  {crab 3067  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530  {csn 4558  ∪ cuni 4836   ↦ cmpt 5153  ⟶wf 6414  ‘cfv 6418  ficfi 9099  Topctop 21950  TopOnctopon 21967  neicnei 22156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-en 8692  df-fin 8695  df-fi 9100  df-top 21951  df-topon 21968  df-ntr 22079  df-nei 22157

This theorem is referenced by:  ustuqtop  23306