Theorem utopval 24207

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 24206	. 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 4864	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → ∪ 𝑢 = ∪ 𝑈)
4	3	dmeqd 5854	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom ∪ 𝑢 = dom ∪ 𝑈)
5		ustbas2 24200	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
6	5	adantr 480	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑋 = dom ∪ 𝑈)
7	4, 6	eqtr4d 2775	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom ∪ 𝑢 = 𝑋)
8	7	pweqd 4559	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝒫 dom ∪ 𝑢 = 𝒫 𝑋)
9	2	rexeqdv 3297	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → (∃𝑣 ∈ 𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎))
10	9	ralbidv 3161	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → (∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎))
11	8, 10	rabeqbidv 3408	. 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 6864	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ ∪ ran UnifOn)
13		elfvex 6869	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
14		pwexg 5315	. . 3 ⊢ (𝑋 ∈ V → 𝒫 𝑋 ∈ V)
15		rabexg 5274	. . 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 6950	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3390  Vcvv 3430   ⊆ wss 3890  𝒫 cpw 4542  {csn 4568  ∪ cuni 4851  dom cdm 5624  ran crn 5625   “ cima 5627  ‘cfv 6492  UnifOncust 24175  unifTopcutop 24205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fv 6500  df-ust 24176  df-utop 24206

This theorem is referenced by:  elutop  24208  utoptop  24209  utopbas  24210  utopsnneiplem  24222  psmetutop  24542