MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tuslem Structured version   Visualization version   GIF version

Theorem tuslem 23991
Description: Lemma for tusbas 23993, tusunif 23994, and tustopn 23996. (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 17151 . . . 4 Base = Slot (Base‘ndx)
2 tsetndxnbasendx 17305 . . . . 5 (TopSet‘ndx) ≠ (Base‘ndx)
32necomi 2995 . . . 4 (Base‘ndx) ≠ (TopSet‘ndx)
41, 3setsnid 17146 . . 3 (Base‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (Base‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
5 ustbas2 23950 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom 𝑈)
6 uniexg 7732 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ V)
7 dmexg 7896 . . . . 5 ( 𝑈 ∈ V → dom 𝑈 ∈ V)
8 eqid 2732 . . . . . 6 {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} = {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}
9 basendxltunifndx 17347 . . . . . 6 (Base‘ndx) < (UnifSet‘ndx)
10 unifndxnn 17346 . . . . . 6 (UnifSet‘ndx) ∈ ℕ
118, 9, 102strbas1 17175 . . . . 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 2772 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
14 tuslem.k . . . . 5 𝐾 = (toUnifSp‘𝑈)
15 tusval 23990 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
1614, 15eqtrid 2784 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 = ({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
1716fveq2d 6895 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (Base‘𝐾) = (Base‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
184, 13, 173eqtr4a 2798 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾))
19 unifid 17345 . . . 4 UnifSet = Slot (UnifSet‘ndx)
20 unifndxntsetndx 17349 . . . 4 (UnifSet‘ndx) ≠ (TopSet‘ndx)
2119, 20setsnid 17146 . . 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 17176 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘{⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
2316fveq2d 6895 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (UnifSet‘𝐾) = (UnifSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
2421, 22, 233eqtr4a 2798 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾))
25 prex 5432 . . . . 5 {⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} ∈ V
26 fvex 6904 . . . . 5 (unifTop‘𝑈) ∈ V
27 tsetid 17302 . . . . . 6 TopSet = Slot (TopSet‘ndx)
2827setsid 17145 . . . . 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 690 . . . 4 (unifTop‘𝑈) = (TopSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
3016fveq2d 6895 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopSet‘({⟨(Base‘ndx), dom 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
3129, 30eqtr4id 2791 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
32 utopbas 23960 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
3331unieqd 4922 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
3432, 18, 333eqtr3rd 2781 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (Base‘𝐾))
3534oveq2d 7427 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → ((TopSet‘𝐾) ↾t (TopSet‘𝐾)) = ((TopSet‘𝐾) ↾t (Base‘𝐾)))
36 fvex 6904 . . . . 5 (TopSet‘𝐾) ∈ V
37 eqid 2732 . . . . . 6 (TopSet‘𝐾) = (TopSet‘𝐾)
3837restid 17383 . . . . 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 2732 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
41 eqid 2732 . . . . 5 (TopSet‘𝐾) = (TopSet‘𝐾)
4240, 41topnval 17384 . . . 4 ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾)
4335, 39, 423eqtr3g 2795 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopOpen‘𝐾))
4431, 43eqtrd 2772 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopOpen‘𝐾))
4518, 24, 443jca 1128 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 1541  wcel 2106  Vcvv 3474  {cpr 4630  cop 4634   cuni 4908  dom cdm 5676  cfv 6543  (class class class)co 7411   sSet csts 17100  ndxcnx 17130  Basecbs 17148  TopSetcts 17207  UnifSetcunif 17211  t crest 17370  TopOpenctopn 17371  UnifOncust 23924  unifTopcutop 23955  toUnifSpctus 23980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13489  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-tset 17220  df-unif 17224  df-rest 17372  df-topn 17373  df-ust 23925  df-utop 23956  df-tus 23983
This theorem is referenced by:  tusbas  23993  tusunif  23994  tustopn  23996  tususp  23997
  Copyright terms: Public domain W3C validator