Theorem tuslem 24296

Description: Lemma for tusbas 24298, tusunif 24299, and tustopn 24301. (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

Step	Hyp	Ref	Expression
1		baseid 17261	. . . 4 ⊢ Base = Slot (Base‘ndx)
2		tsetndxnbasendx 17415	. . . . 5 ⊢ (TopSet‘ndx) ≠ (Base‘ndx)
3	2	necomi 3001	. . . 4 ⊢ (Base‘ndx) ≠ (TopSet‘ndx)
4	1, 3	setsnid 17256	. . 3 ⊢ (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (Base‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
5		ustbas2 24255	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
6		uniexg 7775	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 ∈ V)
7		dmexg 7941	. . . . 5 ⊢ (∪ 𝑈 ∈ V → dom ∪ 𝑈 ∈ V)
8		eqid 2740	. . . . . 6 ⊢ {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} = {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}
9		basendxltunifndx 17457	. . . . . 6 ⊢ (Base‘ndx) < (UnifSet‘ndx)
10		unifndxnn 17456	. . . . . 6 ⊢ (UnifSet‘ndx) ∈ ℕ
11	8, 9, 10	2strbas1 17285	. . . . 5 ⊢ (dom ∪ 𝑈 ∈ V → dom ∪ 𝑈 = (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
12	6, 7, 11	3syl 18	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → dom ∪ 𝑈 = (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
13	5, 12	eqtrd 2780	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
14		tuslem.k	. . . . 5 ⊢ 𝐾 = (toUnifSp‘𝑈)
15		tusval 24295	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
16	14, 15	eqtrid 2792	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 = ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
17	16	fveq2d 6924	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (Base‘𝐾) = (Base‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
18	4, 13, 17	3eqtr4a 2806	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾))
19		unifid 17455	. . . 4 ⊢ UnifSet = Slot (UnifSet‘ndx)
20		unifndxntsetndx 17459	. . . 4 ⊢ (UnifSet‘ndx) ≠ (TopSet‘ndx)
21	19, 20	setsnid 17256	. . 3 ⊢ (UnifSet‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (UnifSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
22	8, 9, 10, 19	2strop1 17286	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
23	16	fveq2d 6924	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (UnifSet‘𝐾) = (UnifSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
24	21, 22, 23	3eqtr4a 2806	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾))
25		prex 5452	. . . . 5 ⊢ {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} ∈ V
26		fvex 6933	. . . . 5 ⊢ (unifTop‘𝑈) ∈ V
27		tsetid 17412	. . . . . 6 ⊢ TopSet = Slot (TopSet‘ndx)
28	27	setsid 17255	. . . . 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)⟩)))
29	25, 26, 28	mp2an 691	. . . 4 ⊢ (unifTop‘𝑈) = (TopSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
30	16	fveq2d 6924	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
31	29, 30	eqtr4id 2799	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
32		utopbas 24265	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))
33	31	unieqd 4944	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ (unifTop‘𝑈) = ∪ (TopSet‘𝐾))
34	32, 18, 33	3eqtr3rd 2789	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ (TopSet‘𝐾) = (Base‘𝐾))
35	34	oveq2d 7464	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((TopSet‘𝐾) ↾t ∪ (TopSet‘𝐾)) = ((TopSet‘𝐾) ↾t (Base‘𝐾)))
36		fvex 6933	. . . . 5 ⊢ (TopSet‘𝐾) ∈ V
37		eqid 2740	. . . . . 6 ⊢ ∪ (TopSet‘𝐾) = ∪ (TopSet‘𝐾)
38	37	restid 17493	. . . . 5 ⊢ ((TopSet‘𝐾) ∈ V → ((TopSet‘𝐾) ↾t ∪ (TopSet‘𝐾)) = (TopSet‘𝐾))
39	36, 38	ax-mp 5	. . . 4 ⊢ ((TopSet‘𝐾) ↾t ∪ (TopSet‘𝐾)) = (TopSet‘𝐾)
40		eqid 2740	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
41		eqid 2740	. . . . 5 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾)
42	40, 41	topnval 17494	. . . 4 ⊢ ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾)
43	35, 39, 42	3eqtr3g 2803	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopOpen‘𝐾))
44	31, 43	eqtrd 2780	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopOpen‘𝐾))
45	18, 24, 44	3jca 1128	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝑈 = (UnifSet‘𝐾) ∧ (unifTop‘𝑈) = (TopOpen‘𝐾)))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  Vcvv 3488  {cpr 4650  ⟨cop 4654  ∪ cuni 4931  dom cdm 5700  ‘cfv 6573  (class class class)co 7448   sSet csts 17210  ndxcnx 17240  Basecbs 17258  TopSetcts 17317  UnifSetcunif 17321   ↾_t crest 17480  TopOpenctopn 17481  UnifOncust 24229  unifTopcutop 24260  toUnifSpctus 24285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-tset 17330  df-unif 17334  df-rest 17482  df-topn 17483  df-ust 24230  df-utop 24261  df-tus 24288

This theorem is referenced by:  tusbas  24298  tusunif  24299  tustopn  24301  tususp  24302