Theorem ustuqtop 22050

Description: For a given uniform structure 𝑈 on a set 𝑋, there is a unique topology 𝑗 such that the set ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})) is the filter of the neighborhoods of 𝑝 for that topology. Proposition 1 of [BourbakiTop1] p. II.3. (Contributed by Thierry Arnoux, 11-Jan-2018.)

Hypothesis

Ref	Expression
utopustuq.1	⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))

Assertion

Ref	Expression
ustuqtop	⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝 ∈ 𝑋 (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝}))

Distinct variable groups:   𝑣,𝑝,𝑈   𝑋,𝑝,𝑣,𝑗   𝑗,𝑁,𝑝   𝑣,𝑗,𝑈   𝑗,𝑋

Allowed substitution hint:   𝑁(𝑣)

Proof of Theorem ustuqtop

Dummy variables 𝑎 𝑏 𝑐 𝑟 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . 7 ⊢ (𝑝 = 𝑟 → (𝑁‘𝑝) = (𝑁‘𝑟))
2	1	eleq2d 2687	. . . . . 6 ⊢ (𝑝 = 𝑟 → (𝑐 ∈ (𝑁‘𝑝) ↔ 𝑐 ∈ (𝑁‘𝑟)))
3	2	cbvralv 3171	. . . . 5 ⊢ (∀𝑝 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑝) ↔ ∀𝑟 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑟))
4		eleq1 2689	. . . . . 6 ⊢ (𝑐 = 𝑎 → (𝑐 ∈ (𝑁‘𝑝) ↔ 𝑎 ∈ (𝑁‘𝑝)))
5	4	raleqbi1dv 3146	. . . . 5 ⊢ (𝑐 = 𝑎 → (∀𝑝 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑝) ↔ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)))
6	3, 5	syl5bbr 274	. . . 4 ⊢ (𝑐 = 𝑎 → (∀𝑟 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑟) ↔ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)))
7	6	cbvrabv 3199	. . 3 ⊢ {𝑐 ∈ 𝒫 𝑋 ∣ ∀𝑟 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑟)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)}
8		utopustuq.1	. . . 4 ⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝})))
9	8	ustuqtop0 22044	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
10	8	ustuqtop1 22045	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))
11	8	ustuqtop2 22046	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))
12	8	ustuqtop3 22047	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)
13	8	ustuqtop4 22048	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑏 ∈ (𝑁‘𝑝)∀𝑥 ∈ 𝑏 𝑎 ∈ (𝑁‘𝑥))
14	8	ustuqtop5 22049	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))
15	7, 9, 10, 11, 12, 13, 14	neiptopreu 20937	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
16	9	feqmptd 6249	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁 = (𝑝 ∈ 𝑋 ↦ (𝑁‘𝑝)))
17	16	eqeq1d 2624	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ (𝑝 ∈ 𝑋 ↦ (𝑁‘𝑝)) = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))))
18		fvex 6201	. . . . . 6 ⊢ (𝑁‘𝑝) ∈ V
19	18	rgenw 2924	. . . . 5 ⊢ ∀𝑝 ∈ 𝑋 (𝑁‘𝑝) ∈ V
20		mpteqb 6299	. . . . 5 ⊢ (∀𝑝 ∈ 𝑋 (𝑁‘𝑝) ∈ V → ((𝑝 ∈ 𝑋 ↦ (𝑁‘𝑝)) = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∀𝑝 ∈ 𝑋 (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝})))
21	19, 20	ax-mp 5	. . . 4 ⊢ ((𝑝 ∈ 𝑋 ↦ (𝑁‘𝑝)) = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∀𝑝 ∈ 𝑋 (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝}))
22	17, 21	syl6bb 276	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∀𝑝 ∈ 𝑋 (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝})))
23	22	reubidv 3126	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝 ∈ 𝑋 (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝})))
24	15, 23	mpbid 222	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝 ∈ 𝑋 (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝}))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃!wreu 2914  {crab 2916  Vcvv 3200  𝒫 cpw 4158  {csn 4177   ↦ cmpt 4729  ran crn 5115   “ cima 5117  ‘cfv 5888  TopOnctopon 20715  neicnei 20901  UnifOncust 22003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-top 20699  df-topon 20716  df-ntr 20824  df-nei 20902  df-ust 22004

This theorem is referenced by: (None)