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Theorem tuslem 21823
Description: Lemma for tusbas 21824, tusunif 21825, and tustopn 21827. (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 15693 . . . 4 Base = Slot (Base‘ndx)
2 1re 9895 . . . . . 6 1 ∈ ℝ
3 1lt9 11076 . . . . . 6 1 < 9
42, 3ltneii 10001 . . . . 5 1 ≠ 9
5 basendx 15697 . . . . . 6 (Base‘ndx) = 1
6 tsetndx 15809 . . . . . 6 (TopSet‘ndx) = 9
75, 6neeq12i 2847 . . . . 5 ((Base‘ndx) ≠ (TopSet‘ndx) ↔ 1 ≠ 9)
84, 7mpbir 219 . . . 4 (Base‘ndx) ≠ (TopSet‘ndx)
91, 8setsnid 15689 . . 3 (Base‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (Base‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
10 ustbas2 21781 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
11 uniexg 6830 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ V)
12 dmexg 6966 . . . . 5 ( 𝑈 ∈ V → dom 𝑈 ∈ V)
13 eqid 2609 . . . . . 6 {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} = {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}
14 df-unif 15738 . . . . . 6 UnifSet = Slot 13
15 1nn 10878 . . . . . . 7 1 ∈ ℕ
16 3nn0 11157 . . . . . . 7 3 ∈ ℕ0
17 1nn0 11155 . . . . . . 7 1 ∈ ℕ0
18 1lt10 11513 . . . . . . 7 1 < 10
1915, 16, 17, 18declti 11378 . . . . . 6 1 < 13
20 3nn 11033 . . . . . . 7 3 ∈ ℕ
2117, 20decnncl 11350 . . . . . 6 13 ∈ ℕ
2213, 14, 19, 212strbas 15756 . . . . 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 2643 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
25 tuslem.k . . . . 5 𝐾 = (toUnifSp‘𝑈)
26 tusval 21822 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
2725, 26syl5eq 2655 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 = ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
2827fveq2d 6092 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (Base‘𝐾) = (Base‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
299, 24, 283eqtr4a 2669 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾))
30 unifid 15834 . . . 4 UnifSet = Slot (UnifSet‘ndx)
31 9re 10954 . . . . . 6 9 ∈ ℝ
32 9nn0 11163 . . . . . . 7 9 ∈ ℕ0
33 9lt10 11505 . . . . . . 7 9 < 10
3415, 16, 32, 33declti 11378 . . . . . 6 9 < 13
3531, 34gtneii 10000 . . . . 5 13 ≠ 9
36 unifndx 15833 . . . . . 6 (UnifSet‘ndx) = 13
3736, 6neeq12i 2847 . . . . 5 ((UnifSet‘ndx) ≠ (TopSet‘ndx) ↔ 13 ≠ 9)
3835, 37mpbir 219 . . . 4 (UnifSet‘ndx) ≠ (TopSet‘ndx)
3930, 38setsnid 15689 . . 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 15757 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
4127fveq2d 6092 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (UnifSet‘𝐾) = (UnifSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
4239, 40, 413eqtr4a 2669 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾))
4327fveq2d 6092 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
44 prex 4831 . . . . 5 {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} ∈ V
45 fvex 6098 . . . . 5 (unifTop‘𝑈) ∈ V
46 tsetid 15810 . . . . . 6 TopSet = Slot (TopSet‘ndx)
4746setsid 15688 . . . . 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 703 . . . 4 (unifTop‘𝑈) = (TopSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
4943, 48syl6reqr 2662 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
50 utopbas 21791 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
5149unieqd 4376 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
5250, 29, 513eqtr3rd 2652 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (Base‘𝐾))
5352oveq2d 6543 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → ((TopSet‘𝐾) ↾t (TopSet‘𝐾)) = ((TopSet‘𝐾) ↾t (Base‘𝐾)))
54 fvex 6098 . . . . 5 (TopSet‘𝐾) ∈ V
55 eqid 2609 . . . . . 6 (TopSet‘𝐾) = (TopSet‘𝐾)
5655restid 15863 . . . . 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 2609 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
59 eqid 2609 . . . . 5 (TopSet‘𝐾) = (TopSet‘𝐾)
6058, 59topnval 15864 . . . 4 ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾)
6153, 57, 603eqtr3g 2666 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopOpen‘𝐾))
6249, 61eqtrd 2643 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopOpen‘𝐾))
6329, 42, 623jca 1234 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 1030   = wceq 1474  wcel 1976  wne 2779  Vcvv 3172  {cpr 4126  cop 4130   cuni 4366  dom cdm 5028  cfv 5790  (class class class)co 6527  1c1 9793  3c3 10918  9c9 10924  cdc 11325  ndxcnx 15638   sSet csts 15639  Basecbs 15641  TopSetcts 15720  UnifSetcunif 15724  t crest 15850  TopOpenctopn 15851  UnifOncust 21755  unifTopcutop 21786  toUnifSpctus 21811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-dec 11326  df-uz 11520  df-fz 12153  df-struct 15643  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-tset 15733  df-unif 15738  df-rest 15852  df-topn 15853  df-ust 21756  df-utop 21787  df-tus 21814
This theorem is referenced by:  tusbas  21824  tusunif  21825  tustopn  21827  tususp  21828
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