Theorem neiptopreu 22637

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 22614, innei 22629, elnei 22615 and neissex 22631, 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 22629 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 9406 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 22635	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Top)
9		toptopon2 22420	. . . 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 22634	. . . 4 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
12	11	fveq2d 6896	. . 3 ⊢ (𝜑 → (TopOn‘𝑋) = (TopOn‘∪ 𝐽))
13	10, 12	eleqtrrd 2837	. 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
14	1, 2, 3, 4, 5, 6, 7	neiptopnei 22636	. 2 ⊢ (𝜑 → 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝐽)‘{𝑝})))
15		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑝(𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋))
16		nfmpt1 5257	. . . . . . . . . . 11 ⊢ Ⅎ𝑝(𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))
17	16	nfeq2 2921	. . . . . . . . . 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 775	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ⊆ 𝑋) ∧ 𝑝 ∈ 𝑏) → 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
22		simpr 486	. . . . . . . . . . . 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 6907	. . . . . . . . . . . 12 ⊢ ((𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ∧ 𝑝 ∈ 𝑋) → ((nei‘𝑗)‘{𝑝}) ∈ V)
26	24, 25	fvmpt2d 7012	. . . . . . . . . . 11 ⊢ ((𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ∧ 𝑝 ∈ 𝑋) → (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝}))
27	21, 23, 26	syl2anc 585	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ⊆ 𝑋) ∧ 𝑝 ∈ 𝑏) → (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝}))
28	27	eqcomd 2739	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ⊆ 𝑋) ∧ 𝑝 ∈ 𝑏) → ((nei‘𝑗)‘{𝑝}) = (𝑁‘𝑝))
29	28	eleq2d 2820	. . . . . . . 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 3268	. . . . . . 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 580	. . . . . 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 22429	. . . . . . . . 9 ⊢ ((𝑗 ∈ (TopOn‘𝑋) ∧ 𝑏 ∈ 𝑗) → 𝑏 ⊆ 𝑋)
33	32	ad4ant24 753	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ∈ 𝑗) → 𝑏 ⊆ 𝑋)
34		topontop 22415	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (TopOn‘𝑋) → 𝑗 ∈ Top)
35	34	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → 𝑗 ∈ Top)
36		opnnei 22624	. . . . . . . . . 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 478	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ∈ 𝑗) → ∀𝑝 ∈ 𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))
39	33, 38	jca 513	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ 𝑏 ∈ 𝑗) → (𝑏 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})))
40	37	biimpar 479	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) ∧ ∀𝑝 ∈ 𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝})) → 𝑏 ∈ 𝑗)
41	40	adantrl 715	. . . . . . 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 800	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → (𝑏 ∈ 𝑗 ↔ (𝑏 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑏 𝑏 ∈ ((nei‘𝑗)‘{𝑝}))))
43	1	neipeltop 22633	. . . . . . 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 2731	. . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) ∧ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))) → 𝑗 = 𝐽)
47	46	ex 414	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (TopOn‘𝑋)) → (𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑗 = 𝐽))
48	47	ralrimiva 3147	. 2 ⊢ (𝜑 → ∀𝑗 ∈ (TopOn‘𝑋)(𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) → 𝑗 = 𝐽))
49		simpl 484	. . . . . . 7 ⊢ ((𝑗 = 𝐽 ∧ 𝑝 ∈ 𝑋) → 𝑗 = 𝐽)
50	49	fveq2d 6896	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑝 ∈ 𝑋) → (nei‘𝑗) = (nei‘𝐽))
51	50	fveq1d 6894	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑝 ∈ 𝑋) → ((nei‘𝑗)‘{𝑝}) = ((nei‘𝐽)‘{𝑝}))
52	51	mpteq2dva 5249	. . . 4 ⊢ (𝑗 = 𝐽 → (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) = (𝑝 ∈ 𝑋 ↦ ((nei‘𝐽)‘{𝑝})))
53	52	eqeq2d 2744	. . 3 ⊢ (𝑗 = 𝐽 → (𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ 𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝐽)‘{𝑝}))))
54	53	eqreu 3726	. 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 1372	1 ⊢ (𝜑 → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  ∃!wreu 3375  {crab 3433  Vcvv 3475   ⊆ wss 3949  𝒫 cpw 4603  {csn 4629  ∪ cuni 4909   ↦ cmpt 5232  ⟶wf 6540  ‘cfv 6544  ficfi 9405  Topctop 22395  TopOnctopon 22412  neicnei 22601

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1o 8466  df-en 8940  df-fin 8943  df-fi 9406  df-top 22396  df-topon 22413  df-ntr 22524  df-nei 22602

This theorem is referenced by:  ustuqtop  23751