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Theorem tuslem 23019

Description: Lemma for tusbas 23020, tusunif 23021, and tustopn 23023. (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**
Step	Hyp	Ref	Expression
1		baseid 16646	. . . 4 ⊢ Base = Slot (Base‘ndx)
2		1re 10719	. . . . . 6 ⊢ 1 ∈ ℝ
3		1lt9 11922	. . . . . 6 ⊢ 1 < 9
4	2, 3	ltneii 10831	. . . . 5 ⊢ 1 ≠ 9
5		basendx 16650	. . . . . 6 ⊢ (Base‘ndx) = 1
6		tsetndx 16762	. . . . . 6 ⊢ (TopSet‘ndx) = 9
7	5, 6	neeq12i 3000	. . . . 5 ⊢ ((Base‘ndx) ≠ (TopSet‘ndx) ↔ 1 ≠ 9)
8	4, 7	mpbir 234	. . . 4 ⊢ (Base‘ndx) ≠ (TopSet‘ndx)
9	1, 8	setsnid 16642	. . 3 ⊢ (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (Base‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
10		ustbas2 22977	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
11		uniexg 7484	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 ∈ V)
12		dmexg 7634	. . . . 5 ⊢ (∪ 𝑈 ∈ V → dom ∪ 𝑈 ∈ V)
13		eqid 2738	. . . . . 6 ⊢ {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} = {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}
14		df-unif 16691	. . . . . 6 ⊢ UnifSet = Slot ;13
15		1nn 11727	. . . . . . 7 ⊢ 1 ∈ ℕ
16		3nn0 11994	. . . . . . 7 ⊢ 3 ∈ ℕ₀
17		1nn0 11992	. . . . . . 7 ⊢ 1 ∈ ℕ₀
18		1lt10 12318	. . . . . . 7 ⊢ 1 < ;10
19	15, 16, 17, 18	declti 12217	. . . . . 6 ⊢ 1 < ;13
20		3nn 11795	. . . . . . 7 ⊢ 3 ∈ ℕ
21	17, 20	decnncl 12199	. . . . . 6 ⊢ ;13 ∈ ℕ
22	13, 14, 19, 21	2strbas 16706	. . . . 5 ⊢ (dom ∪ 𝑈 ∈ V → dom ∪ 𝑈 = (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
23	11, 12, 22	3syl 18	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → dom ∪ 𝑈 = (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
24	10, 23	eqtrd 2773	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
25		tuslem.k	. . . . 5 ⊢ 𝐾 = (toUnifSp‘𝑈)
26		tusval 23018	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
27	25, 26	syl5eq 2785	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 = ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
28	27	fveq2d 6678	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (Base‘𝐾) = (Base‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
29	9, 24, 28	3eqtr4a 2799	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾))
30		unifid 16781	. . . 4 ⊢ UnifSet = Slot (UnifSet‘ndx)
31		9re 11815	. . . . . 6 ⊢ 9 ∈ ℝ
32		9nn0 12000	. . . . . . 7 ⊢ 9 ∈ ℕ₀
33		9lt10 12310	. . . . . . 7 ⊢ 9 < ;10
34	15, 16, 32, 33	declti 12217	. . . . . 6 ⊢ 9 < ;13
35	31, 34	gtneii 10830	. . . . 5 ⊢ ;13 ≠ 9
36		unifndx 16780	. . . . . 6 ⊢ (UnifSet‘ndx) = ;13
37	36, 6	neeq12i 3000	. . . . 5 ⊢ ((UnifSet‘ndx) ≠ (TopSet‘ndx) ↔ ;13 ≠ 9)
38	35, 37	mpbir 234	. . . 4 ⊢ (UnifSet‘ndx) ≠ (TopSet‘ndx)
39	30, 38	setsnid 16642	. . 3 ⊢ (UnifSet‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (UnifSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
40	13, 14, 19, 21	2strop 16707	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
41	27	fveq2d 6678	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (UnifSet‘𝐾) = (UnifSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
42	39, 40, 41	3eqtr4a 2799	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾))
43		prex 5299	. . . . 5 ⊢ {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} ∈ V
44		fvex 6687	. . . . 5 ⊢ (unifTop‘𝑈) ∈ V
45		tsetid 16763	. . . . . 6 ⊢ TopSet = Slot (TopSet‘ndx)
46	45	setsid 16641	. . . . 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)⟩)))
47	43, 44, 46	mp2an 692	. . . 4 ⊢ (unifTop‘𝑈) = (TopSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
48	27	fveq2d 6678	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
49	47, 48	eqtr4id 2792	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
50		utopbas 22987	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))
51	49	unieqd 4810	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ (unifTop‘𝑈) = ∪ (TopSet‘𝐾))
52	50, 29, 51	3eqtr3rd 2782	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ (TopSet‘𝐾) = (Base‘𝐾))
53	52	oveq2d 7186	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((TopSet‘𝐾) ↾_t ∪ (TopSet‘𝐾)) = ((TopSet‘𝐾) ↾_t (Base‘𝐾)))
54		fvex 6687	. . . . 5 ⊢ (TopSet‘𝐾) ∈ V
55		eqid 2738	. . . . . 6 ⊢ ∪ (TopSet‘𝐾) = ∪ (TopSet‘𝐾)
56	55	restid 16810	. . . . 5 ⊢ ((TopSet‘𝐾) ∈ V → ((TopSet‘𝐾) ↾_t ∪ (TopSet‘𝐾)) = (TopSet‘𝐾))
57	54, 56	ax-mp 5	. . . 4 ⊢ ((TopSet‘𝐾) ↾_t ∪ (TopSet‘𝐾)) = (TopSet‘𝐾)
58		eqid 2738	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
59		eqid 2738	. . . . 5 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾)
60	58, 59	topnval 16811	. . . 4 ⊢ ((TopSet‘𝐾) ↾_t (Base‘𝐾)) = (TopOpen‘𝐾)
61	53, 57, 60	3eqtr3g 2796	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopOpen‘𝐾))
62	49, 61	eqtrd 2773	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopOpen‘𝐾))
63	29, 42, 62	3jca 1129	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 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 Vcvv 3398 {cpr 4518 ⟨cop 4522 ∪ cuni 4796 dom cdm 5525 ‘cfv 6339 (class class class)co 7170 1c1 10616 3c3 11772 9c9 11778 cdc 12179 ndxcnx 16583 sSet csts 16584 Basecbs 16586 TopSetcts 16674 UnifSetcunif 16678 ↾_t crest 16797 TopOpenctopn 16798 UnifOncust 22951 unifTopcutop 22982 toUnifSpctus 23007

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-tset 16687 df-unif 16691 df-rest 16799 df-topn 16800 df-ust 22952 df-utop 22983 df-tus 23010

This theorem is referenced by: tusbas 23020 tusunif 23021 tustopn 23023 tususp 23024

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