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Theorem tuslem 24275
Description: Lemma for tusbas 24277, tusunif 24278, and tustopn 24280. (Contributed by Thierry Arnoux, 5-Dec-2017.) (Proof shortened by AV, 28-Oct-2024.)
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 17250 . . . 4 Base = Slot (Base‘ndx)
2 tsetndxnbasendx 17400 . . . . 5 (TopSet‘ndx) ≠ (Base‘ndx)
32necomi 2995 . . . 4 (Base‘ndx) ≠ (TopSet‘ndx)
41, 3setsnid 17245 . . 3 (Base‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (Base‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
5 ustbas2 24234 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
6 uniexg 7760 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ V)
7 dmexg 7923 . . . . 5 ( 𝑈 ∈ V → dom 𝑈 ∈ V)
8 eqid 2737 . . . . . 6 {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} = {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}
9 basendxltunifndx 17442 . . . . . 6 (Base‘ndx) < (UnifSet‘ndx)
10 unifndxnn 17441 . . . . . 6 (UnifSet‘ndx) ∈ ℕ
118, 9, 102strbas1 17272 . . . . 5 (dom 𝑈 ∈ V → dom 𝑈 = (Base‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
126, 7, 113syl 18 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → dom 𝑈 = (Base‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
135, 12eqtrd 2777 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
14 tuslem.k . . . . 5 𝐾 = (toUnifSp‘𝑈)
15 tusval 24274 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
1614, 15eqtrid 2789 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 = ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
1716fveq2d 6910 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (Base‘𝐾) = (Base‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
184, 13, 173eqtr4a 2803 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾))
19 unifid 17440 . . . 4 UnifSet = Slot (UnifSet‘ndx)
20 unifndxntsetndx 17444 . . . 4 (UnifSet‘ndx) ≠ (TopSet‘ndx)
2119, 20setsnid 17245 . . 3 (UnifSet‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (UnifSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
228, 9, 10, 192strop1 17273 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
2316fveq2d 6910 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (UnifSet‘𝐾) = (UnifSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
2421, 22, 233eqtr4a 2803 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾))
25 prex 5437 . . . . 5 {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} ∈ V
26 fvex 6919 . . . . 5 (unifTop‘𝑈) ∈ V
27 tsetid 17397 . . . . . 6 TopSet = Slot (TopSet‘ndx)
2827setsid 17244 . . . . 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)⟩)))
2925, 26, 28mp2an 692 . . . 4 (unifTop‘𝑈) = (TopSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
3016fveq2d 6910 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
3129, 30eqtr4id 2796 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
32 utopbas 24244 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
3331unieqd 4920 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
3432, 18, 333eqtr3rd 2786 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (Base‘𝐾))
3534oveq2d 7447 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → ((TopSet‘𝐾) ↾t (TopSet‘𝐾)) = ((TopSet‘𝐾) ↾t (Base‘𝐾)))
36 fvex 6919 . . . . 5 (TopSet‘𝐾) ∈ V
37 eqid 2737 . . . . . 6 (TopSet‘𝐾) = (TopSet‘𝐾)
3837restid 17478 . . . . 5 ((TopSet‘𝐾) ∈ V → ((TopSet‘𝐾) ↾t (TopSet‘𝐾)) = (TopSet‘𝐾))
3936, 38ax-mp 5 . . . 4 ((TopSet‘𝐾) ↾t (TopSet‘𝐾)) = (TopSet‘𝐾)
40 eqid 2737 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
41 eqid 2737 . . . . 5 (TopSet‘𝐾) = (TopSet‘𝐾)
4240, 41topnval 17479 . . . 4 ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾)
4335, 39, 423eqtr3g 2800 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopOpen‘𝐾))
4431, 43eqtrd 2777 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopOpen‘𝐾))
4518, 24, 443jca 1129 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 1087   = wceq 1540  wcel 2108  Vcvv 3480  {cpr 4628  cop 4632   cuni 4907  dom cdm 5685  cfv 6561  (class class class)co 7431   sSet csts 17200  ndxcnx 17230  Basecbs 17247  TopSetcts 17303  UnifSetcunif 17307  t crest 17465  TopOpenctopn 17466  UnifOncust 24208  unifTopcutop 24239  toUnifSpctus 24264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-tset 17316  df-unif 17320  df-rest 17467  df-topn 17468  df-ust 24209  df-utop 24240  df-tus 24267
This theorem is referenced by:  tusbas  24277  tusunif  24278  tustopn  24280  tususp  24281
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