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Theorem utopval 22208
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 22207 . . 3 unifTop = (𝑢 ran UnifOn ↦ {𝑎 ∈ 𝒫 dom 𝑢 ∣ ∀𝑥𝑎𝑣𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎})
21a1i 11 . 2 (𝑈 ∈ (UnifOn‘𝑋) → unifTop = (𝑢 ran UnifOn ↦ {𝑎 ∈ 𝒫 dom 𝑢 ∣ ∀𝑥𝑎𝑣𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎}))
3 simpr 479 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
43unieqd 4586 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
54dmeqd 5469 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom 𝑢 = dom 𝑈)
6 ustbas2 22201 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
76adantr 472 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝑋 = dom 𝑈)
85, 7eqtr4d 2785 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → dom 𝑢 = 𝑋)
98pweqd 4295 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → 𝒫 dom 𝑢 = 𝒫 𝑋)
103rexeqdv 3272 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → (∃𝑣𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∃𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎))
1110ralbidv 3112 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → (∀𝑥𝑎𝑣𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎 ↔ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎))
129, 11rabeqbidv 3323 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 = 𝑈) → {𝑎 ∈ 𝒫 dom 𝑢 ∣ ∀𝑥𝑎𝑣𝑢 (𝑣 “ {𝑥}) ⊆ 𝑎} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
13 elrnust 22200 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ran UnifOn)
14 elfvex 6370 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
15 pwexg 4987 . . 3 (𝑋 ∈ V → 𝒫 𝑋 ∈ V)
16 rabexg 4951 . . 3 (𝒫 𝑋 ∈ V → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ∈ V)
1714, 15, 163syl 18 . 2 (𝑈 ∈ (UnifOn‘𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎} ∈ V)
182, 12, 13, 17fvmptd 6438 1 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑥𝑎𝑣𝑈 (𝑣 “ {𝑥}) ⊆ 𝑎})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1620  wcel 2127  wral 3038  wrex 3039  {crab 3042  Vcvv 3328  wss 3703  𝒫 cpw 4290  {csn 4309   cuni 4576  cmpt 4869  dom cdm 5254  ran crn 5255  cima 5257  cfv 6037  UnifOncust 22175  unifTopcutop 22206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-iota 6000  df-fun 6039  df-fn 6040  df-fv 6045  df-ust 22176  df-utop 22207
This theorem is referenced by:  elutop  22209  utoptop  22210  utopbas  22211  utopsnneiplem  22223  psmetutop  22544
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