Theorem ustuqtop 22857

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 6672	. . . . . . 7 ⊢ (𝑝 = 𝑟 → (𝑁‘𝑝) = (𝑁‘𝑟))
2	1	eleq2d 2900	. . . . . 6 ⊢ (𝑝 = 𝑟 → (𝑐 ∈ (𝑁‘𝑝) ↔ 𝑐 ∈ (𝑁‘𝑟)))
3	2	cbvralvw 3451	. . . . 5 ⊢ (∀𝑝 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑝) ↔ ∀𝑟 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑟))
4		eleq1w 2897	. . . . . 6 ⊢ (𝑐 = 𝑎 → (𝑐 ∈ (𝑁‘𝑝) ↔ 𝑎 ∈ (𝑁‘𝑝)))
5	4	raleqbi1dv 3405	. . . . 5 ⊢ (𝑐 = 𝑎 → (∀𝑝 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑝) ↔ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)))
6	3, 5	syl5bbr 287	. . . 4 ⊢ (𝑐 = 𝑎 → (∀𝑟 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑟) ↔ ∀𝑝 ∈ 𝑎 𝑎 ∈ (𝑁‘𝑝)))
7	6	cbvrabv 3493	. . 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 22851	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
10	8	ustuqtop1 22852	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))
11	8	ustuqtop2 22853	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))
12	8	ustuqtop3 22854	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)
13	8	ustuqtop4 22855	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑏 ∈ (𝑁‘𝑝)∀𝑥 ∈ 𝑏 𝑎 ∈ (𝑁‘𝑥))
14	8	ustuqtop5 22856	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))
15	7, 9, 10, 11, 12, 13, 14	neiptopreu 21743	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
16	9	feqmptd 6735	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁 = (𝑝 ∈ 𝑋 ↦ (𝑁‘𝑝)))
17	16	eqeq1d 2825	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ (𝑝 ∈ 𝑋 ↦ (𝑁‘𝑝)) = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))))
18		fvex 6685	. . . . . 6 ⊢ (𝑁‘𝑝) ∈ V
19	18	rgenw 3152	. . . . 5 ⊢ ∀𝑝 ∈ 𝑋 (𝑁‘𝑝) ∈ V
20		mpteqb 6789	. . . . 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 289	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ ∀𝑝 ∈ 𝑋 (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝})))
23	22	reubidv 3391	. 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 234	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)∀𝑝 ∈ 𝑋 (𝑁‘𝑝) = ((nei‘𝑗)‘{𝑝}))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃!wreu 3142  {crab 3144  Vcvv 3496  𝒫 cpw 4541  {csn 4569   ↦ cmpt 5148  ran crn 5558   “ cima 5560  ‘cfv 6357  TopOnctopon 21520  neicnei 21707  UnifOncust 22810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-en 8512  df-fin 8515  df-fi 8877  df-top 21504  df-topon 21521  df-ntr 21630  df-nei 21708  df-ust 22811

This theorem is referenced by: (None)