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Theorem isusp 22797
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 3510 . 2 (𝑊 ∈ UnifSp → 𝑊 ∈ V)
2 0nep0 5249 . . . . 5 ∅ ≠ {∅}
3 isusp.1 . . . . . . . . . . . 12 𝐵 = (Base‘𝑊)
4 fvprc 6656 . . . . . . . . . . . 12 𝑊 ∈ V → (Base‘𝑊) = ∅)
53, 4syl5eq 2865 . . . . . . . . . . 11 𝑊 ∈ V → 𝐵 = ∅)
65fveq2d 6667 . . . . . . . . . 10 𝑊 ∈ V → (UnifOn‘𝐵) = (UnifOn‘∅))
7 ust0 22755 . . . . . . . . . 10 (UnifOn‘∅) = {{∅}}
86, 7syl6eq 2869 . . . . . . . . 9 𝑊 ∈ V → (UnifOn‘𝐵) = {{∅}})
98eleq2d 2895 . . . . . . . 8 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 ∈ {{∅}}))
10 isusp.2 . . . . . . . . . 10 𝑈 = (UnifSt‘𝑊)
1110fvexi 6677 . . . . . . . . 9 𝑈 ∈ V
1211elsn 4572 . . . . . . . 8 (𝑈 ∈ {{∅}} ↔ 𝑈 = {∅})
139, 12syl6bb 288 . . . . . . 7 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 = {∅}))
14 fvprc 6656 . . . . . . . . 9 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
1510, 14syl5eq 2865 . . . . . . . 8 𝑊 ∈ V → 𝑈 = ∅)
1615eqeq1d 2820 . . . . . . 7 𝑊 ∈ V → (𝑈 = {∅} ↔ ∅ = {∅}))
1713, 16bitrd 280 . . . . . 6 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ = {∅}))
1817necon3bbid 3050 . . . . 5 𝑊 ∈ V → (¬ 𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ ≠ {∅}))
192, 18mpbiri 259 . . . 4 𝑊 ∈ V → ¬ 𝑈 ∈ (UnifOn‘𝐵))
2019con4i 114 . . 3 (𝑈 ∈ (UnifOn‘𝐵) → 𝑊 ∈ V)
2120adantr 481 . 2 ((𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)) → 𝑊 ∈ V)
22 fveq2 6663 . . . . . 6 (𝑤 = 𝑊 → (UnifSt‘𝑤) = (UnifSt‘𝑊))
2322, 10syl6eqr 2871 . . . . 5 (𝑤 = 𝑊 → (UnifSt‘𝑤) = 𝑈)
24 fveq2 6663 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
2524, 3syl6eqr 2871 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
2625fveq2d 6667 . . . . 5 (𝑤 = 𝑊 → (UnifOn‘(Base‘𝑤)) = (UnifOn‘𝐵))
2723, 26eleq12d 2904 . . . 4 (𝑤 = 𝑊 → ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ↔ 𝑈 ∈ (UnifOn‘𝐵)))
28 fveq2 6663 . . . . . 6 (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
29 isusp.3 . . . . . 6 𝐽 = (TopOpen‘𝑊)
3028, 29syl6eqr 2871 . . . . 5 (𝑤 = 𝑊 → (TopOpen‘𝑤) = 𝐽)
3123fveq2d 6667 . . . . 5 (𝑤 = 𝑊 → (unifTop‘(UnifSt‘𝑤)) = (unifTop‘𝑈))
3230, 31eqeq12d 2834 . . . 4 (𝑤 = 𝑊 → ((TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)) ↔ 𝐽 = (unifTop‘𝑈)))
3327, 32anbi12d 630 . . 3 (𝑤 = 𝑊 → (((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤))) ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
34 df-usp 22793 . . 3 UnifSp = {𝑤 ∣ ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)))}
3533, 34elab2g 3665 . 2 (𝑊 ∈ V → (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
361, 21, 35pm5.21nii 380 1 (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  Vcvv 3492  c0 4288  {csn 4557  cfv 6348  Basecbs 16471  TopOpenctopn 16683  UnifOncust 22735  unifTopcutop 22766  UnifStcuss 22789  UnifSpcusp 22790
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-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-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-ust 22736  df-usp 22793
This theorem is referenced by:  ressust  22800  ressusp  22801  tususp  22808  uspreg  22810  ucncn  22821  neipcfilu  22832  ucnextcn  22840  xmsusp  23106
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