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Theorem tuslem 21881
Description: Lemma for tusbas 21882, tusunif 21883, and tustopn 21885. (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 15747 . . . 4 Base = Slot (Base‘ndx)
2 1re 9918 . . . . . 6 1 ∈ ℝ
3 1lt9 11106 . . . . . 6 1 < 9
42, 3ltneii 10029 . . . . 5 1 ≠ 9
5 basendx 15751 . . . . . 6 (Base‘ndx) = 1
6 tsetndx 15863 . . . . . 6 (TopSet‘ndx) = 9
75, 6neeq12i 2848 . . . . 5 ((Base‘ndx) ≠ (TopSet‘ndx) ↔ 1 ≠ 9)
84, 7mpbir 220 . . . 4 (Base‘ndx) ≠ (TopSet‘ndx)
91, 8setsnid 15743 . . 3 (Base‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (Base‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
10 ustbas2 21839 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
11 uniexg 6853 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ V)
12 dmexg 6989 . . . . 5 ( 𝑈 ∈ V → dom 𝑈 ∈ V)
13 eqid 2610 . . . . . 6 {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} = {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}
14 df-unif 15792 . . . . . 6 UnifSet = Slot 13
15 1nn 10908 . . . . . . 7 1 ∈ ℕ
16 3nn0 11187 . . . . . . 7 3 ∈ ℕ0
17 1nn0 11185 . . . . . . 7 1 ∈ ℕ0
18 1lt10 11557 . . . . . . 7 1 < 10
1915, 16, 17, 18declti 11422 . . . . . 6 1 < 13
20 3nn 11063 . . . . . . 7 3 ∈ ℕ
2117, 20decnncl 11394 . . . . . 6 13 ∈ ℕ
2213, 14, 19, 212strbas 15810 . . . . 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 2644 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
25 tuslem.k . . . . 5 𝐾 = (toUnifSp‘𝑈)
26 tusval 21880 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
2725, 26syl5eq 2656 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 = ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
2827fveq2d 6107 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (Base‘𝐾) = (Base‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
299, 24, 283eqtr4a 2670 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾))
30 unifid 15888 . . . 4 UnifSet = Slot (UnifSet‘ndx)
31 9re 10984 . . . . . 6 9 ∈ ℝ
32 9nn0 11193 . . . . . . 7 9 ∈ ℕ0
33 9lt10 11549 . . . . . . 7 9 < 10
3415, 16, 32, 33declti 11422 . . . . . 6 9 < 13
3531, 34gtneii 10028 . . . . 5 13 ≠ 9
36 unifndx 15887 . . . . . 6 (UnifSet‘ndx) = 13
3736, 6neeq12i 2848 . . . . 5 ((UnifSet‘ndx) ≠ (TopSet‘ndx) ↔ 13 ≠ 9)
3835, 37mpbir 220 . . . 4 (UnifSet‘ndx) ≠ (TopSet‘ndx)
3930, 38setsnid 15743 . . 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 15811 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
4127fveq2d 6107 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (UnifSet‘𝐾) = (UnifSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
4239, 40, 413eqtr4a 2670 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾))
4327fveq2d 6107 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
44 prex 4836 . . . . 5 {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} ∈ V
45 fvex 6113 . . . . 5 (unifTop‘𝑈) ∈ V
46 tsetid 15864 . . . . . 6 TopSet = Slot (TopSet‘ndx)
4746setsid 15742 . . . . 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 704 . . . 4 (unifTop‘𝑈) = (TopSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
4943, 48syl6reqr 2663 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
50 utopbas 21849 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
5149unieqd 4382 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
5250, 29, 513eqtr3rd 2653 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (Base‘𝐾))
5352oveq2d 6565 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → ((TopSet‘𝐾) ↾t (TopSet‘𝐾)) = ((TopSet‘𝐾) ↾t (Base‘𝐾)))
54 fvex 6113 . . . . 5 (TopSet‘𝐾) ∈ V
55 eqid 2610 . . . . . 6 (TopSet‘𝐾) = (TopSet‘𝐾)
5655restid 15917 . . . . 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 2610 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
59 eqid 2610 . . . . 5 (TopSet‘𝐾) = (TopSet‘𝐾)
6058, 59topnval 15918 . . . 4 ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾)
6153, 57, 603eqtr3g 2667 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopOpen‘𝐾))
6249, 61eqtrd 2644 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopOpen‘𝐾))
6329, 42, 623jca 1235 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 1031   = wceq 1475  wcel 1977  wne 2780  Vcvv 3173  {cpr 4127  cop 4131   cuni 4372  dom cdm 5038  cfv 5804  (class class class)co 6549  1c1 9816  3c3 10948  9c9 10954  cdc 11369  ndxcnx 15692   sSet csts 15693  Basecbs 15695  TopSetcts 15774  UnifSetcunif 15778  t crest 15904  TopOpenctopn 15905  UnifOncust 21813  unifTopcutop 21844  toUnifSpctus 21869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-tset 15787  df-unif 15792  df-rest 15906  df-topn 15907  df-ust 21814  df-utop 21845  df-tus 21872
This theorem is referenced by:  tusbas  21882  tusunif  21883  tustopn  21885  tususp  21886
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