Theorem ustuqtop 24595

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 6908	. . . . . . 7 ⊢ (𝑝 = 𝑟 → (𝑁‘𝑝) = (𝑁‘𝑟))
2	1	eleq2d 2850	. . . . . 6 ⊢ (𝑝 = 𝑟 → (𝑐 ∈ (𝑁‘𝑝) ↔ 𝑐 ∈ (𝑁‘𝑟)))
3	2	cbvralvw 3244	. . . . 5 ⊢ (∀𝑝 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑝) ↔ ∀𝑟 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑟))
4		eleq1w 2847	. . . . . 6 ⊢ (𝑐 = 𝑎 → (𝑐 ∈ (𝑁‘𝑝) ↔ 𝑎 ∈ (𝑁‘𝑝)))
5	4	raleqbi1dv 3333	. . . . 5 ⊢ (𝑐 = 𝑎 → (∀𝑝 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑝) ↔ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)))
6	3, 5	bitr3id 288	. . . 4 ⊢ (𝑐 = 𝑎 → (∀𝑟 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑟) ↔ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)))
7	6	cbvrabv 3426	. . 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 24589	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
10	8	ustuqtop1 24590	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))
11	8	ustuqtop2 24591	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))
12	8	ustuqtop3 24592	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)
13	8	ustuqtop4 24593	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑏 ∈ (𝑁‘𝑝)∀𝑥 ∈ 𝑏 𝑎 ∈ (𝑁‘𝑥))
14	8	ustuqtop5 24594	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))
15	7, 9, 10, 11, 12, 13, 14	neiptopreu 23481	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
16	9	feqmptd 6976	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁 = (𝑝 ∈ 𝑋 ↦ (𝑁‘𝑝)))
17	16	eqeq1d 2765	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ (𝑝 ∈ 𝑋 ↦ (𝑁‘𝑝)) = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))))
18		fvex 6921	. . . . . 6 ⊢ (𝑁‘𝑝) ∈ V
19	18	rgenw 3084	. . . . 5 ⊢ ∀𝑝 ∈ 𝑋 (𝑁‘𝑝) ∈ V
20		mpteqb 7036	. . . . 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	bitrdi 290	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∀𝑝 ∈ 𝑋 (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝})))
23	22	reubidv 3385	. 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 235	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝 ∈ 𝑋 (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃!wreu 3367  {crab 3416  Vcvv 3454  𝒫 cpw 4560  {csn 4588   ↦ cmpt 5189  ran crn 5656   “ cima 5658  ‘cfv 6558  TopOnctopon 23258  neicnei 23445  UnifOncust 24549

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-om 7902  df-1o 8504  df-2o 8505  df-en 9002  df-fin 9005  df-fi 9430  df-top 23242  df-topon 23259  df-ntr 23368  df-nei 23446  df-ust 24550

This theorem is used by: (None)