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Theorem isusp 22059
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 3210 . 2 (𝑊 ∈ UnifSp → 𝑊 ∈ V)
2 0nep0 4834 . . . . 5 ∅ ≠ {∅}
3 isusp.1 . . . . . . . . . . . 12 𝐵 = (Base‘𝑊)
4 fvprc 6183 . . . . . . . . . . . 12 𝑊 ∈ V → (Base‘𝑊) = ∅)
53, 4syl5eq 2667 . . . . . . . . . . 11 𝑊 ∈ V → 𝐵 = ∅)
65fveq2d 6193 . . . . . . . . . 10 𝑊 ∈ V → (UnifOn‘𝐵) = (UnifOn‘∅))
7 ust0 22017 . . . . . . . . . 10 (UnifOn‘∅) = {{∅}}
86, 7syl6eq 2671 . . . . . . . . 9 𝑊 ∈ V → (UnifOn‘𝐵) = {{∅}})
98eleq2d 2686 . . . . . . . 8 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 ∈ {{∅}}))
10 isusp.2 . . . . . . . . . 10 𝑈 = (UnifSt‘𝑊)
11 fvex 6199 . . . . . . . . . 10 (UnifSt‘𝑊) ∈ V
1210, 11eqeltri 2696 . . . . . . . . 9 𝑈 ∈ V
1312elsn 4190 . . . . . . . 8 (𝑈 ∈ {{∅}} ↔ 𝑈 = {∅})
149, 13syl6bb 276 . . . . . . 7 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 = {∅}))
15 fvprc 6183 . . . . . . . . 9 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
1610, 15syl5eq 2667 . . . . . . . 8 𝑊 ∈ V → 𝑈 = ∅)
1716eqeq1d 2623 . . . . . . 7 𝑊 ∈ V → (𝑈 = {∅} ↔ ∅ = {∅}))
1814, 17bitrd 268 . . . . . 6 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ = {∅}))
1918necon3bbid 2830 . . . . 5 𝑊 ∈ V → (¬ 𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ ≠ {∅}))
202, 19mpbiri 248 . . . 4 𝑊 ∈ V → ¬ 𝑈 ∈ (UnifOn‘𝐵))
2120con4i 113 . . 3 (𝑈 ∈ (UnifOn‘𝐵) → 𝑊 ∈ V)
2221adantr 481 . 2 ((𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)) → 𝑊 ∈ V)
23 fveq2 6189 . . . . . 6 (𝑤 = 𝑊 → (UnifSt‘𝑤) = (UnifSt‘𝑊))
2423, 10syl6eqr 2673 . . . . 5 (𝑤 = 𝑊 → (UnifSt‘𝑤) = 𝑈)
25 fveq2 6189 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
2625, 3syl6eqr 2673 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
2726fveq2d 6193 . . . . 5 (𝑤 = 𝑊 → (UnifOn‘(Base‘𝑤)) = (UnifOn‘𝐵))
2824, 27eleq12d 2694 . . . 4 (𝑤 = 𝑊 → ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ↔ 𝑈 ∈ (UnifOn‘𝐵)))
29 fveq2 6189 . . . . . 6 (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
30 isusp.3 . . . . . 6 𝐽 = (TopOpen‘𝑊)
3129, 30syl6eqr 2673 . . . . 5 (𝑤 = 𝑊 → (TopOpen‘𝑤) = 𝐽)
3224fveq2d 6193 . . . . 5 (𝑤 = 𝑊 → (unifTop‘(UnifSt‘𝑤)) = (unifTop‘𝑈))
3331, 32eqeq12d 2636 . . . 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 22055 . . 3 UnifSp = {𝑤 ∣ ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)))}
3634, 35elab2g 3351 . 2 (𝑊 ∈ V → (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
371, 22, 36pm5.21nii 368 1 (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 384   = wceq 1482  wcel 1989  wne 2793  Vcvv 3198  c0 3913  {csn 4175  cfv 5886  Basecbs 15851  TopOpenctopn 16076  UnifOncust 21997  unifTopcutop 22028  UnifStcuss 22051  UnifSpcusp 22052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-res 5124  df-iota 5849  df-fun 5888  df-fv 5894  df-ust 21998  df-usp 22055
This theorem is referenced by:  ressust  22062  ressusp  22063  tususp  22070  uspreg  22072  ucncn  22083  neipcfilu  22094  ucnextcn  22102  xmsusp  22368
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