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Theorem tuslem 22118
 Description: Lemma for tusbas 22119, tusunif 22120, and tustopn 22122. (Contributed by Thierry Arnoux, 5-Dec-2017.)
Hypothesis
Ref Expression
tuslem.k 𝐾 = (toUnifSp‘𝑈)
Assertion
Ref Expression
tuslem (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝑈 = (UnifSet‘𝐾) ∧ (unifTop‘𝑈) = (TopOpen‘𝐾)))

Proof of Theorem tuslem
StepHypRef Expression
1 baseid 15966 . . . 4 Base = Slot (Base‘ndx)
2 1re 10077 . . . . . 6 1 ∈ ℝ
3 1lt9 11267 . . . . . 6 1 < 9
42, 3ltneii 10188 . . . . 5 1 ≠ 9
5 basendx 15970 . . . . . 6 (Base‘ndx) = 1
6 tsetndx 16087 . . . . . 6 (TopSet‘ndx) = 9
75, 6neeq12i 2889 . . . . 5 ((Base‘ndx) ≠ (TopSet‘ndx) ↔ 1 ≠ 9)
84, 7mpbir 221 . . . 4 (Base‘ndx) ≠ (TopSet‘ndx)
91, 8setsnid 15962 . . 3 (Base‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (Base‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
10 ustbas2 22076 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
11 uniexg 6997 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ V)
12 dmexg 7139 . . . . 5 ( 𝑈 ∈ V → dom 𝑈 ∈ V)
13 eqid 2651 . . . . . 6 {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} = {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}
14 df-unif 16012 . . . . . 6 UnifSet = Slot 13
15 1nn 11069 . . . . . . 7 1 ∈ ℕ
16 3nn0 11348 . . . . . . 7 3 ∈ ℕ0
17 1nn0 11346 . . . . . . 7 1 ∈ ℕ0
18 1lt10 11719 . . . . . . 7 1 < 10
1915, 16, 17, 18declti 11584 . . . . . 6 1 < 13
20 3nn 11224 . . . . . . 7 3 ∈ ℕ
2117, 20decnncl 11556 . . . . . 6 13 ∈ ℕ
2213, 14, 19, 212strbas 16031 . . . . 5 (dom 𝑈 ∈ V → dom 𝑈 = (Base‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
2311, 12, 223syl 18 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → dom 𝑈 = (Base‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
2410, 23eqtrd 2685 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
25 tuslem.k . . . . 5 𝐾 = (toUnifSp‘𝑈)
26 tusval 22117 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
2725, 26syl5eq 2697 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 = ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
2827fveq2d 6233 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (Base‘𝐾) = (Base‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
299, 24, 283eqtr4a 2711 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾))
30 unifid 16112 . . . 4 UnifSet = Slot (UnifSet‘ndx)
31 9re 11145 . . . . . 6 9 ∈ ℝ
32 9nn0 11354 . . . . . . 7 9 ∈ ℕ0
33 9lt10 11711 . . . . . . 7 9 < 10
3415, 16, 32, 33declti 11584 . . . . . 6 9 < 13
3531, 34gtneii 10187 . . . . 5 13 ≠ 9
36 unifndx 16111 . . . . . 6 (UnifSet‘ndx) = 13
3736, 6neeq12i 2889 . . . . 5 ((UnifSet‘ndx) ≠ (TopSet‘ndx) ↔ 13 ≠ 9)
3835, 37mpbir 221 . . . 4 (UnifSet‘ndx) ≠ (TopSet‘ndx)
3930, 38setsnid 15962 . . 3 (UnifSet‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (UnifSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
4013, 14, 19, 212strop 16032 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
4127fveq2d 6233 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (UnifSet‘𝐾) = (UnifSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
4239, 40, 413eqtr4a 2711 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾))
4327fveq2d 6233 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
44 prex 4939 . . . . 5 {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} ∈ V
45 fvex 6239 . . . . 5 (unifTop‘𝑈) ∈ V
46 tsetid 16088 . . . . . 6 TopSet = Slot (TopSet‘ndx)
4746setsid 15961 . . . . 5 (({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} ∈ V ∧ (unifTop‘𝑈) ∈ V) → (unifTop‘𝑈) = (TopSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
4844, 45, 47mp2an 708 . . . 4 (unifTop‘𝑈) = (TopSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
4943, 48syl6reqr 2704 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
50 utopbas 22086 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
5149unieqd 4478 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
5250, 29, 513eqtr3rd 2694 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (Base‘𝐾))
5352oveq2d 6706 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → ((TopSet‘𝐾) ↾t (TopSet‘𝐾)) = ((TopSet‘𝐾) ↾t (Base‘𝐾)))
54 fvex 6239 . . . . 5 (TopSet‘𝐾) ∈ V
55 eqid 2651 . . . . . 6 (TopSet‘𝐾) = (TopSet‘𝐾)
5655restid 16141 . . . . 5 ((TopSet‘𝐾) ∈ V → ((TopSet‘𝐾) ↾t (TopSet‘𝐾)) = (TopSet‘𝐾))
5754, 56ax-mp 5 . . . 4 ((TopSet‘𝐾) ↾t (TopSet‘𝐾)) = (TopSet‘𝐾)
58 eqid 2651 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
59 eqid 2651 . . . . 5 (TopSet‘𝐾) = (TopSet‘𝐾)
6058, 59topnval 16142 . . . 4 ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾)
6153, 57, 603eqtr3g 2708 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopOpen‘𝐾))
6249, 61eqtrd 2685 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopOpen‘𝐾))
6329, 42, 623jca 1261 1 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝑈 = (UnifSet‘𝐾) ∧ (unifTop‘𝑈) = (TopOpen‘𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  Vcvv 3231  {cpr 4212  ⟨cop 4216  ∪ cuni 4468  dom cdm 5143  ‘cfv 5926  (class class class)co 6690  1c1 9975  3c3 11109  9c9 11115  ;cdc 11531  ndxcnx 15901   sSet csts 15902  Basecbs 15904  TopSetcts 15994  UnifSetcunif 15998   ↾t crest 16128  TopOpenctopn 16129  UnifOncust 22050  unifTopcutop 22081  toUnifSpctus 22106 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-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  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-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  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-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-tset 16007  df-unif 16012  df-rest 16130  df-topn 16131  df-ust 22051  df-utop 22082  df-tus 22109 This theorem is referenced by:  tusbas  22119  tusunif  22120  tustopn  22122  tususp  22123
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