Theorem neiptopreu 23141

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 23118, innei 23133, elnei 23119 and neissex 23135, 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 23133 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 9451 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 23139	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Top)
9		toptopon2 22924	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
10	8, 9	sylib 218	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘∪ 𝐽))
11	1, 2, 3, 4, 5, 6, 7	neiptopuni 23138	. . . 4 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
12	11	fveq2d 6910	. . 3 ⊢ (𝜑 → (TopOn‘𝑋) = (TopOn‘∪ 𝐽))
13	10, 12	eleqtrrd 2844	. 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
14	1, 2, 3, 4, 5, 6, 7	neiptopnei 23140	. 2 ⊢ (𝜑 → 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝐽)‘{𝑝})))
15		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑝(𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋))
16		nfmpt1 5250	. . . . . . . . . . 11 ⊢ Ⅎ𝑝(𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))
17	16	nfeq2 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑝 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))
18	15, 17	nfan 1899	. . . . . . . . 9 ⊢ Ⅎ𝑝((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
19		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑝 𝑏 ⊆ 𝑋
20	18, 19	nfan 1899	. . . . . . . 8 ⊢ Ⅎ𝑝(((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ⊆ 𝑋)
21		simpllr 776	. . . . . . . . . . 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 3983	. . . . . . . . . . 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 6921	. . . . . . . . . . . 12 ⊢ ((𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ∧ 𝑝 ∈ 𝑋) → ((nei‘𝑗)‘{𝑝}) ∈ V)
26	24, 25	fvmpt2d 7029	. . . . . . . . . . 11 ⊢ ((𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ∧ 𝑝 ∈ 𝑋) → (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝}))
27	21, 23, 26	syl2anc 584	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ⊆ 𝑋) ∧ 𝑝 ∈ 𝑏) → (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝}))
28	27	eqcomd 2743	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ⊆ 𝑋) ∧ 𝑝 ∈ 𝑏) → ((nei‘𝑗)‘{𝑝}) = (𝑁‘𝑝))
29	28	eleq2d 2827	. . . . . . . 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 3270	. . . . . . 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 579	. . . . . 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 22933	. . . . . . . . 9 ⊢ ((𝑗 ∈ (TopOn‘𝑋) ∧ 𝑏 ∈ 𝑗) → 𝑏 ⊆ 𝑋)
33	32	ad4ant24 754	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ∈ 𝑗) → 𝑏 ⊆ 𝑋)
34		topontop 22919	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (TopOn‘𝑋) → 𝑗 ∈ Top)
35	34	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → 𝑗 ∈ Top)
36		opnnei 23128	. . . . . . . . . 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 716	. . . . . . 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 801	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏 ∈ 𝑗 ↔ (𝑏 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))))
43	1	neipeltop 23137	. . . . . . 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 311	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏 ∈ 𝑗 ↔ 𝑏 ∈ 𝐽))
46	45	eqrdv 2735	. . . 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 3146	. 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 6910	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑝 ∈ 𝑋) → (nei‘𝑗) = (nei‘𝐽))
51	50	fveq1d 6908	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑝 ∈ 𝑋) → ((nei‘𝑗)‘{𝑝}) = ((nei‘𝐽)‘{𝑝}))
52	51	mpteq2dva 5242	. . . 4 ⊢ (𝑗 = 𝐽 → (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) = (𝑝 ∈ 𝑋 ↦ ((nei‘𝐽)‘{𝑝})))
53	52	eqeq2d 2748	. . 3 ⊢ (𝑗 = 𝐽 → (𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝐽)‘{𝑝}))))
54	53	eqreu 3735	. 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 1373	1 ⊢ (𝜑 → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  ∃!wreu 3378  {crab 3436  Vcvv 3480   ⊆ wss 3951  𝒫 cpw 4600  {csn 4626  ∪ cuni 4907   ↦ cmpt 5225  ⟶wf 6557  ‘cfv 6561  ficfi 9450  Topctop 22899  TopOnctopon 22916  neicnei 23105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-2o 8507  df-en 8986  df-fin 8989  df-fi 9451  df-top 22900  df-topon 22917  df-ntr 23028  df-nei 23106

This theorem is referenced by:  ustuqtop  24255