MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  utopval Structured version   Visualization version   GIF version

Theorem utopval 23729
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 ∣ ∀𝑥𝑎𝑣𝑈 (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.
StepHypRef Expression
1 df-utop 23728 . 2 unifTop = (𝑢 ran UnifOn ↦ {𝑎 ∈ 𝒫 dom 𝑢 ∣ ∀𝑥𝑎𝑣𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎})
2 simpr 486 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
32unieqd 4922 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
43dmeqd 5904 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom 𝑢 = dom 𝑈)
5 ustbas2 23722 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
65adantr 482 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑋 = dom 𝑈)
74, 6eqtr4d 2776 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom 𝑢 = 𝑋)
87pweqd 4619 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝒫 dom 𝑢 = 𝒫 𝑋)
92rexeqdv 3327 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → (∃𝑣𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎))
109ralbidv 3178 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → (∀𝑥𝑎𝑣𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎))
118, 10rabeqbidv 3450 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → {𝑎 ∈ 𝒫 dom 𝑢 ∣ ∀𝑥𝑎𝑣𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
12 elfvunirn 6921 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
13 elfvex 6927 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
14 pwexg 5376 . . 3 (𝑋 ∈ V → 𝒫 𝑋 ∈ V)
15 rabexg 5331 . . 3 (𝒫 𝑋 ∈ V → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ∈ V)
1613, 14, 153syl 18 . 2 (𝑈 ∈ (UnifOn‘𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ∈ V)
171, 11, 12, 16fvmptd2 7004 1 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  {crab 3433  Vcvv 3475  wss 3948  𝒫 cpw 4602  {csn 4628   cuni 4908  dom cdm 5676  ran crn 5677  cima 5679  cfv 6541  UnifOncust 23696  unifTopcutop 23727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6493  df-fun 6543  df-fv 6549  df-ust 23697  df-utop 23728
This theorem is referenced by:  elutop  23730  utoptop  23731  utopbas  23732  utopsnneiplem  23744  psmetutop  24068
  Copyright terms: Public domain W3C validator