Theorem utopval 24354

Description: The topology induced by a uniform structure 𝑈. (Contributed by Thierry Arnoux, 30-Nov-2017.)

Assertion

Ref	Expression
utopval	⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})

Distinct variable groups:   𝑣,𝑎,𝑥,𝑈   𝑋,𝑎,𝑥

Allowed substitution hint:   𝑋(𝑣)

Proof of Theorem utopval

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-utop 24353	. 2 ⊢ unifTop = (𝑢 ∈ ∪ ran UnifOn ↦ {𝑎 ∈ 𝒫 dom ∪ 𝑢 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎})
2		simpr 489	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
3	2	unieqd 4886	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ∪ 𝑢 = ∪ 𝑈)
4	3	dmeqd 5893	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom ∪ 𝑢 = dom ∪ 𝑈)
5		ustbas2 24347	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
6	5	adantr 485	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑋 = dom ∪ 𝑈)
7	4, 6	eqtr4d 2807	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom ∪ 𝑢 = 𝑋)
8	7	pweqd 4581	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝒫 dom ∪ 𝑢 = 𝒫 𝑋)
9	2	rexeqdv 3330	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → (∃𝑣 ∈ 𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎))
10	9	ralbidv 3194	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → (∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎))
11	8, 10	rabeqbidv 3441	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → {𝑎 ∈ 𝒫 dom ∪ 𝑢 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
12		elfvunirn 6909	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)
13		elfvex 6914	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
14		pwexg 5347	. . 3 ⊢ (𝑋 ∈ V → 𝒫 𝑋 ∈ V)
15		rabexg 5305	. . 3 ⊢ (𝒫 𝑋 ∈ V → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ∈ V)
16	13, 14, 15	3syl 19	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ∈ V)
17	1, 11, 12, 16	fvmptd2 6996	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  {crab 3423  Vcvv 3463   ⊆ wss 3913  𝒫 cpw 4564  {csn 4591  ∪ cuni 4873  dom cdm 5659  ran crn 5660   “ cima 5662  ‘cfv 6533  UnifOncust 24322  unifTopcutop 24352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6489  df-fun 6535  df-fv 6541  df-ust 24323  df-utop 24353

This theorem is referenced by:  elutop  24355  utoptop  24356  utopbas  24357  utopsnneiplem  24369  psmetutop  24689