Theorem ustuqtop 24382

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 6881	. . . . . . 7 ⊢ (𝑝 = 𝑟 → (𝑁‘𝑝) = (𝑁‘𝑟))
2	1	eleq2d 2847	. . . . . 6 ⊢ (𝑝 = 𝑟 → (𝑐 ∈ (𝑁‘𝑝) ↔ 𝑐 ∈ (𝑁‘𝑟)))
3	2	cbvralvw 3241	. . . . 5 ⊢ (∀𝑝 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑝) ↔ ∀𝑟 ∈ 𝑐 𝑐 ∈ (𝑁‘𝑟))
4		eleq1w 2844	. . . . . 6 ⊢ (𝑐 = 𝑎 → (𝑐 ∈ (𝑁‘𝑝) ↔ 𝑎 ∈ (𝑁‘𝑝)))
5	4	raleqbi1dv 3331	. . . . 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 3424	. . 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 24376	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁:𝑋⟶𝒫 𝒫 𝑋)
10	8	ustuqtop1 24377	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ⊆ 𝑏 ∧ 𝑏 ⊆ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑏 ∈ (𝑁‘𝑝))
11	8	ustuqtop2 24378	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝))
12	8	ustuqtop3 24379	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → 𝑝 ∈ 𝑎)
13	8	ustuqtop4 24380	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑏 ∈ (𝑁‘𝑝)∀𝑥 ∈ 𝑏 𝑎 ∈ (𝑁‘𝑥))
14	8	ustuqtop5 24381	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ (𝑁‘𝑝))
15	7, 9, 10, 11, 12, 13, 14	neiptopreu 23269	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∃!𝑗 ∈ (TopOn‘𝑋)𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})))
16	9	feqmptd 6949	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑁 = (𝑝 ∈ 𝑋 ↦ (𝑁‘𝑝)))
17	16	eqeq1d 2763	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑁 = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝})) ↔ (𝑝 ∈ 𝑋 ↦ (𝑁‘𝑝)) = (𝑝 ∈ 𝑋 ↦ ((nei‘𝑗)‘{𝑝}))))
18		fvex 6894	. . . . . 6 ⊢ (𝑁‘𝑝) ∈ V
19	18	rgenw 3081	. . . . 5 ⊢ ∀𝑝 ∈ 𝑋 (𝑁‘𝑝) ∈ V
20		mpteqb 7009	. . . . 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 3383	. 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}))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1568   ∈ wcel 2141  ∀wral 3077  ∃!wreu 3365  {crab 3414  Vcvv 3453  𝒫 cpw 4561  {csn 4588   ↦ cmpt 5191  ran crn 5662   “ cima 5664  ‘cfv 6536  TopOnctopon 23046  neicnei 23233  UnifOncust 24336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7862  df-1o 8452  df-2o 8453  df-en 8943  df-fin 8946  df-fi 9370  df-top 23030  df-topon 23047  df-ntr 23156  df-nei 23234  df-ust 24337

This theorem is referenced by: (None)