Theorem tuslem 24565

Description: Lemma for tusbas 24566, tusunif 24567, and tustopn 24569. (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 17370	. . . 4 ⊢ Base = Slot (Base‘ndx)
2		tsetndxnbasendx 17507	. . . . 5 ⊢ (TopSet‘ndx) ≠ (Base‘ndx)
3	2	necomi 3010	. . . 4 ⊢ (Base‘ndx) ≠ (TopSet‘ndx)
4	1, 3	setsnid 17366	. . 3 ⊢ (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (Base‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
5		ustbas2 24524	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
6		uniexg 7746	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 ∈ V)
7		dmexg 7902	. . . . 5 ⊢ (∪ 𝑈 ∈ V → dom ∪ 𝑈 ∈ V)
8		eqid 2761	. . . . . 6 ⊢ {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} = {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}
9		basendxltunifndx 17549	. . . . . 6 ⊢ (Base‘ndx) < (UnifSet‘ndx)
10		unifndxnn 17548	. . . . . 6 ⊢ (UnifSet‘ndx) ∈ ℕ
11	8, 9, 10	2strbas 17386	. . . . 5 ⊢ (dom ∪ 𝑈 ∈ V → dom ∪ 𝑈 = (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
12	6, 7, 11	3syl 19	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → dom ∪ 𝑈 = (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
13	5, 12	eqtrd 2796	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
14		tuslem.k	. . . . 5 ⊢ 𝐾 = (toUnifSp‘𝑈)
15		tusval 24564	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
16	14, 15	eqtrid 2808	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 = ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
17	16	fveq2d 6881	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (Base‘𝐾) = (Base‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
18	4, 13, 17	3eqtr4a 2822	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾))
19		unifid 17547	. . . 4 ⊢ UnifSet = Slot (UnifSet‘ndx)
20		unifndxntsetndx 17551	. . . 4 ⊢ (UnifSet‘ndx) ≠ (TopSet‘ndx)
21	19, 20	setsnid 17366	. . 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 17387	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
23	16	fveq2d 6881	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (UnifSet‘𝐾) = (UnifSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
24	21, 22, 23	3eqtr4a 2822	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾))
25		prex 5396	. . . . 5 ⊢ {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} ∈ V
26		fvex 6890	. . . . 5 ⊢ (unifTop‘𝑈) ∈ V
27		tsetid 17504	. . . . . 6 ⊢ TopSet = Slot (TopSet‘ndx)
28	27	setsid 17365	. . . . 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 705	. . . 4 ⊢ (unifTop‘𝑈) = (TopSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
30	16	fveq2d 6881	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
31	29, 30	eqtr4id 2815	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
32		utopbas 24534	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))
33	31	unieqd 4880	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ (unifTop‘𝑈) = ∪ (TopSet‘𝐾))
34	32, 18, 33	3eqtr3rd 2805	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ (TopSet‘𝐾) = (Base‘𝐾))
35	34	oveq2d 7428	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((TopSet‘𝐾) ↾t ∪ (TopSet‘𝐾)) = ((TopSet‘𝐾) ↾t (Base‘𝐾)))
36		fvex 6890	. . . . 5 ⊢ (TopSet‘𝐾) ∈ V
37		eqid 2761	. . . . . 6 ⊢ ∪ (TopSet‘𝐾) = ∪ (TopSet‘𝐾)
38	37	restid 17584	. . . . 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 2761	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
41		eqid 2761	. . . . 5 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾)
42	40, 41	topnval 17585	. . . 4 ⊢ ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾)
43	35, 39, 42	3eqtr3g 2819	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopOpen‘𝐾))
44	31, 43	eqtrd 2796	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopOpen‘𝐾))
45	18, 24, 44	3jca 1146	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝑈 = (UnifSet‘𝐾) ∧ (unifTop‘𝑈) = (TopOpen‘𝐾)))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3451  {cpr 4586  ⟨cop 4590  ∪ cuni 4867  dom cdm 5651  ‘cfv 6531  (class class class)co 7412   sSet csts 17321  ndxcnx 17351  Basecbs 17367  TopSetcts 17414  UnifSetcunif 17418   ↾_t crest 17571  TopOpenctopn 17572  UnifOncust 24499  unifTopcutop 24529  toUnifSpctus 24554

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-er 8701  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-3 12387  df-4 12388  df-5 12389  df-6 12390  df-7 12391  df-8 12392  df-9 12393  df-n0 12588  df-z 12675  df-dec 12796  df-uz 12947  df-fz 13621  df-struct 17305  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-tset 17427  df-unif 17431  df-rest 17573  df-topn 17574  df-ust 24500  df-utop 24530  df-tus 24557

This theorem is used by:  tusbas  24566  tusunif  24567  tustopn  24569  tususp  24570