Theorem tuslem 24154

Description: Lemma for tusbas 24155, tusunif 24156, and tustopn 24158. (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 17182	. . . 4 ⊢ Base = Slot (Base‘ndx)
2		tsetndxnbasendx 17319	. . . . 5 ⊢ (TopSet‘ndx) ≠ (Base‘ndx)
3	2	necomi 2979	. . . 4 ⊢ (Base‘ndx) ≠ (TopSet‘ndx)
4	1, 3	setsnid 17178	. . 3 ⊢ (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (Base‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
5		ustbas2 24113	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
6		uniexg 7716	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 ∈ V)
7		dmexg 7877	. . . . 5 ⊢ (∪ 𝑈 ∈ V → dom ∪ 𝑈 ∈ V)
8		eqid 2729	. . . . . 6 ⊢ {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} = {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}
9		basendxltunifndx 17361	. . . . . 6 ⊢ (Base‘ndx) < (UnifSet‘ndx)
10		unifndxnn 17360	. . . . . 6 ⊢ (UnifSet‘ndx) ∈ ℕ
11	8, 9, 10	2strbas 17198	. . . . 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 2764	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
14		tuslem.k	. . . . 5 ⊢ 𝐾 = (toUnifSp‘𝑈)
15		tusval 24153	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
16	14, 15	eqtrid 2776	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 = ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
17	16	fveq2d 6862	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (Base‘𝐾) = (Base‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
18	4, 13, 17	3eqtr4a 2790	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾))
19		unifid 17359	. . . 4 ⊢ UnifSet = Slot (UnifSet‘ndx)
20		unifndxntsetndx 17363	. . . 4 ⊢ (UnifSet‘ndx) ≠ (TopSet‘ndx)
21	19, 20	setsnid 17178	. . 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	2strop 17199	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
23	16	fveq2d 6862	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (UnifSet‘𝐾) = (UnifSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
24	21, 22, 23	3eqtr4a 2790	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾))
25		prex 5392	. . . . 5 ⊢ {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} ∈ V
26		fvex 6871	. . . . 5 ⊢ (unifTop‘𝑈) ∈ V
27		tsetid 17316	. . . . . 6 ⊢ TopSet = Slot (TopSet‘ndx)
28	27	setsid 17177	. . . . 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 692	. . . 4 ⊢ (unifTop‘𝑈) = (TopSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
30	16	fveq2d 6862	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
31	29, 30	eqtr4id 2783	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
32		utopbas 24123	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))
33	31	unieqd 4884	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ (unifTop‘𝑈) = ∪ (TopSet‘𝐾))
34	32, 18, 33	3eqtr3rd 2773	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ (TopSet‘𝐾) = (Base‘𝐾))
35	34	oveq2d 7403	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((TopSet‘𝐾) ↾t ∪ (TopSet‘𝐾)) = ((TopSet‘𝐾) ↾t (Base‘𝐾)))
36		fvex 6871	. . . . 5 ⊢ (TopSet‘𝐾) ∈ V
37		eqid 2729	. . . . . 6 ⊢ ∪ (TopSet‘𝐾) = ∪ (TopSet‘𝐾)
38	37	restid 17396	. . . . 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 2729	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
41		eqid 2729	. . . . 5 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾)
42	40, 41	topnval 17397	. . . 4 ⊢ ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾)
43	35, 39, 42	3eqtr3g 2787	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopOpen‘𝐾))
44	31, 43	eqtrd 2764	. 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 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3447  {cpr 4591  ⟨cop 4595  ∪ cuni 4871  dom cdm 5638  ‘cfv 6511  (class class class)co 7387   sSet csts 17133  ndxcnx 17163  Basecbs 17179  TopSetcts 17226  UnifSetcunif 17230   ↾_t crest 17383  TopOpenctopn 17384  UnifOncust 24087  unifTopcutop 24118  toUnifSpctus 24143

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-fz 13469  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-tset 17239  df-unif 17243  df-rest 17385  df-topn 17386  df-ust 24088  df-utop 24119  df-tus 24146

This theorem is referenced by:  tusbas  24155  tusunif  24156  tustopn  24158  tususp  24159