Theorem utopval 24256

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 24255	. 2 ⊢ unifTop = (𝑢 ∈ ∪ ran UnifOn ↦ {𝑎 ∈ 𝒫 dom ∪ 𝑢 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎})
2		simpr 484	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
3	2	unieqd 4924	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ∪ 𝑢 = ∪ 𝑈)
4	3	dmeqd 5918	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom ∪ 𝑢 = dom ∪ 𝑈)
5		ustbas2 24249	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
6	5	adantr 480	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑋 = dom ∪ 𝑈)
7	4, 6	eqtr4d 2777	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom ∪ 𝑢 = 𝑋)
8	7	pweqd 4621	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝒫 dom ∪ 𝑢 = 𝒫 𝑋)
9	2	rexeqdv 3324	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → (∃𝑣 ∈ 𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎))
10	9	ralbidv 3175	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → (∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎))
11	8, 10	rabeqbidv 3451	. 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 6938	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)
13		elfvex 6944	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
14		pwexg 5383	. . 3 ⊢ (𝑋 ∈ V → 𝒫 𝑋 ∈ V)
15		rabexg 5342	. . 3 ⊢ (𝒫 𝑋 ∈ V → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ∈ V)
16	13, 14, 15	3syl 18	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ∈ V)
17	1, 11, 12, 16	fvmptd2 7023	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067  {crab 3432  Vcvv 3477   ⊆ wss 3962  𝒫 cpw 4604  {csn 4630  ∪ cuni 4911  dom cdm 5688  ran crn 5689   “ cima 5691  ‘cfv 6562  UnifOncust 24223  unifTopcutop 24254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-iota 6515  df-fun 6564  df-fv 6570  df-ust 24224  df-utop 24255

This theorem is referenced by:  elutop  24257  utoptop  24258  utopbas  24259  utopsnneiplem  24271  psmetutop  24595