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Theorem utopval 22768
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 22767 . 2 unifTop = (𝑢 ran UnifOn ↦ {𝑎 ∈ 𝒫 dom 𝑢 ∣ ∀𝑥𝑎𝑣𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎})
2 simpr 485 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
32unieqd 4840 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
43dmeqd 5767 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom 𝑢 = dom 𝑈)
5 ustbas2 22761 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
65adantr 481 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑋 = dom 𝑈)
74, 6eqtr4d 2856 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom 𝑢 = 𝑋)
87pweqd 4540 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝒫 dom 𝑢 = 𝒫 𝑋)
92rexeqdv 3414 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → (∃𝑣𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎))
109ralbidv 3194 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → (∀𝑥𝑎𝑣𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎))
118, 10rabeqbidv 3483 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → {𝑎 ∈ 𝒫 dom 𝑢 ∣ ∀𝑥𝑎𝑣𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
12 elrnust 22760 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
13 elfvex 6696 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
14 pwexg 5270 . . 3 (𝑋 ∈ V → 𝒫 𝑋 ∈ V)
15 rabexg 5225 . . 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 6768 1 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  wss 3933  𝒫 cpw 4535  {csn 4557   cuni 4830  dom cdm 5548  ran crn 5549  cima 5551  cfv 6348  UnifOncust 22735  unifTopcutop 22766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-ust 22736  df-utop 22767
This theorem is referenced by:  elutop  22769  utoptop  22770  utopbas  22771  utopsnneiplem  22783  psmetutop  23104
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