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Theorem isusp 22112
Description: The predicate 𝑊 is a uniform space. (Contributed by Thierry Arnoux, 4-Dec-2017.)
Hypotheses
Ref Expression
isusp.1 𝐵 = (Base‘𝑊)
isusp.2 𝑈 = (UnifSt‘𝑊)
isusp.3 𝐽 = (TopOpen‘𝑊)
Assertion
Ref Expression
isusp (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))

Proof of Theorem isusp
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elex 3243 . 2 (𝑊 ∈ UnifSp → 𝑊 ∈ V)
2 0nep0 4866 . . . . 5 ∅ ≠ {∅}
3 isusp.1 . . . . . . . . . . . 12 𝐵 = (Base‘𝑊)
4 fvprc 6223 . . . . . . . . . . . 12 𝑊 ∈ V → (Base‘𝑊) = ∅)
53, 4syl5eq 2697 . . . . . . . . . . 11 𝑊 ∈ V → 𝐵 = ∅)
65fveq2d 6233 . . . . . . . . . 10 𝑊 ∈ V → (UnifOn‘𝐵) = (UnifOn‘∅))
7 ust0 22070 . . . . . . . . . 10 (UnifOn‘∅) = {{∅}}
86, 7syl6eq 2701 . . . . . . . . 9 𝑊 ∈ V → (UnifOn‘𝐵) = {{∅}})
98eleq2d 2716 . . . . . . . 8 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 ∈ {{∅}}))
10 isusp.2 . . . . . . . . . 10 𝑈 = (UnifSt‘𝑊)
11 fvex 6239 . . . . . . . . . 10 (UnifSt‘𝑊) ∈ V
1210, 11eqeltri 2726 . . . . . . . . 9 𝑈 ∈ V
1312elsn 4225 . . . . . . . 8 (𝑈 ∈ {{∅}} ↔ 𝑈 = {∅})
149, 13syl6bb 276 . . . . . . 7 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 = {∅}))
15 fvprc 6223 . . . . . . . . 9 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
1610, 15syl5eq 2697 . . . . . . . 8 𝑊 ∈ V → 𝑈 = ∅)
1716eqeq1d 2653 . . . . . . 7 𝑊 ∈ V → (𝑈 = {∅} ↔ ∅ = {∅}))
1814, 17bitrd 268 . . . . . 6 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ = {∅}))
1918necon3bbid 2860 . . . . 5 𝑊 ∈ V → (¬ 𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ ≠ {∅}))
202, 19mpbiri 248 . . . 4 𝑊 ∈ V → ¬ 𝑈 ∈ (UnifOn‘𝐵))
2120con4i 113 . . 3 (𝑈 ∈ (UnifOn‘𝐵) → 𝑊 ∈ V)
2221adantr 480 . 2 ((𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)) → 𝑊 ∈ V)
23 fveq2 6229 . . . . . 6 (𝑤 = 𝑊 → (UnifSt‘𝑤) = (UnifSt‘𝑊))
2423, 10syl6eqr 2703 . . . . 5 (𝑤 = 𝑊 → (UnifSt‘𝑤) = 𝑈)
25 fveq2 6229 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
2625, 3syl6eqr 2703 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
2726fveq2d 6233 . . . . 5 (𝑤 = 𝑊 → (UnifOn‘(Base‘𝑤)) = (UnifOn‘𝐵))
2824, 27eleq12d 2724 . . . 4 (𝑤 = 𝑊 → ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ↔ 𝑈 ∈ (UnifOn‘𝐵)))
29 fveq2 6229 . . . . . 6 (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
30 isusp.3 . . . . . 6 𝐽 = (TopOpen‘𝑊)
3129, 30syl6eqr 2703 . . . . 5 (𝑤 = 𝑊 → (TopOpen‘𝑤) = 𝐽)
3224fveq2d 6233 . . . . 5 (𝑤 = 𝑊 → (unifTop‘(UnifSt‘𝑤)) = (unifTop‘𝑈))
3331, 32eqeq12d 2666 . . . 4 (𝑤 = 𝑊 → ((TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)) ↔ 𝐽 = (unifTop‘𝑈)))
3428, 33anbi12d 747 . . 3 (𝑤 = 𝑊 → (((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤))) ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
35 df-usp 22108 . . 3 UnifSp = {𝑤 ∣ ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)))}
3634, 35elab2g 3385 . 2 (𝑊 ∈ V → (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
371, 22, 36pm5.21nii 367 1 (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  Vcvv 3231  c0 3948  {csn 4210  cfv 5926  Basecbs 15904  TopOpenctopn 16129  UnifOncust 22050  unifTopcutop 22081  UnifStcuss 22104  UnifSpcusp 22105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934  df-ust 22051  df-usp 22108
This theorem is referenced by:  ressust  22115  ressusp  22116  tususp  22123  uspreg  22125  ucncn  22136  neipcfilu  22147  ucnextcn  22155  xmsusp  22421
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