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Theorem tuslem 22071

Description: Lemma for tusbas 22072, tusunif 22073, and tustopn 22075. (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 15919	. . . 4 ⊢ Base = Slot (Base‘ndx)
2		1re 10039	. . . . . 6 ⊢ 1 ∈ ℝ
3		1lt9 11229	. . . . . 6 ⊢ 1 < 9
4	2, 3	ltneii 10150	. . . . 5 ⊢ 1 ≠ 9
5		basendx 15923	. . . . . 6 ⊢ (Base‘ndx) = 1
6		tsetndx 16040	. . . . . 6 ⊢ (TopSet‘ndx) = 9
7	5, 6	neeq12i 2860	. . . . 5 ⊢ ((Base‘ndx) ≠ (TopSet‘ndx) ↔ 1 ≠ 9)
8	4, 7	mpbir 221	. . . 4 ⊢ (Base‘ndx) ≠ (TopSet‘ndx)
9	1, 8	setsnid 15915	. . 3 ⊢ (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (Base‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
10		ustbas2 22029	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
11		uniexg 6955	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 ∈ V)
12		dmexg 7097	. . . . 5 ⊢ (∪ 𝑈 ∈ V → dom ∪ 𝑈 ∈ V)
13		eqid 2622	. . . . . 6 ⊢ {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} = {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}
14		df-unif 15965	. . . . . 6 ⊢ UnifSet = Slot ;13
15		1nn 11031	. . . . . . 7 ⊢ 1 ∈ ℕ
16		3nn0 11310	. . . . . . 7 ⊢ 3 ∈ ℕ₀
17		1nn0 11308	. . . . . . 7 ⊢ 1 ∈ ℕ₀
18		1lt10 11681	. . . . . . 7 ⊢ 1 < ;10
19	15, 16, 17, 18	declti 11546	. . . . . 6 ⊢ 1 < ;13
20		3nn 11186	. . . . . . 7 ⊢ 3 ∈ ℕ
21	17, 20	decnncl 11518	. . . . . 6 ⊢ ;13 ∈ ℕ
22	13, 14, 19, 21	2strbas 15984	. . . . 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 2656	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
25		tuslem.k	. . . . 5 ⊢ 𝐾 = (toUnifSp‘𝑈)
26		tusval 22070	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
27	25, 26	syl5eq 2668	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 = ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
28	27	fveq2d 6195	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (Base‘𝐾) = (Base‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
29	9, 24, 28	3eqtr4a 2682	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾))
30		unifid 16065	. . . 4 ⊢ UnifSet = Slot (UnifSet‘ndx)
31		9re 11107	. . . . . 6 ⊢ 9 ∈ ℝ
32		9nn0 11316	. . . . . . 7 ⊢ 9 ∈ ℕ₀
33		9lt10 11673	. . . . . . 7 ⊢ 9 < ;10
34	15, 16, 32, 33	declti 11546	. . . . . 6 ⊢ 9 < ;13
35	31, 34	gtneii 10149	. . . . 5 ⊢ ;13 ≠ 9
36		unifndx 16064	. . . . . 6 ⊢ (UnifSet‘ndx) = ;13
37	36, 6	neeq12i 2860	. . . . 5 ⊢ ((UnifSet‘ndx) ≠ (TopSet‘ndx) ↔ ;13 ≠ 9)
38	35, 37	mpbir 221	. . . 4 ⊢ (UnifSet‘ndx) ≠ (TopSet‘ndx)
39	30, 38	setsnid 15915	. . 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 15985	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
41	27	fveq2d 6195	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (UnifSet‘𝐾) = (UnifSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
42	39, 40, 41	3eqtr4a 2682	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾))
43	27	fveq2d 6195	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
44		prex 4909	. . . . 5 ⊢ {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} ∈ V
45		fvex 6201	. . . . 5 ⊢ (unifTop‘𝑈) ∈ V
46		tsetid 16041	. . . . . 6 ⊢ TopSet = Slot (TopSet‘ndx)
47	46	setsid 15914	. . . . 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)⟩)))
48	44, 45, 47	mp2an 708	. . . 4 ⊢ (unifTop‘𝑈) = (TopSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
49	43, 48	syl6reqr 2675	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
50		utopbas 22039	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))
51	49	unieqd 4446	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ (unifTop‘𝑈) = ∪ (TopSet‘𝐾))
52	50, 29, 51	3eqtr3rd 2665	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ (TopSet‘𝐾) = (Base‘𝐾))
53	52	oveq2d 6666	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((TopSet‘𝐾) ↾_t ∪ (TopSet‘𝐾)) = ((TopSet‘𝐾) ↾_t (Base‘𝐾)))
54		fvex 6201	. . . . 5 ⊢ (TopSet‘𝐾) ∈ V
55		eqid 2622	. . . . . 6 ⊢ ∪ (TopSet‘𝐾) = ∪ (TopSet‘𝐾)
56	55	restid 16094	. . . . 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 2622	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
59		eqid 2622	. . . . 5 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾)
60	58, 59	topnval 16095	. . . 4 ⊢ ((TopSet‘𝐾) ↾_t (Base‘𝐾)) = (TopOpen‘𝐾)
61	53, 57, 60	3eqtr3g 2679	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopOpen‘𝐾))
62	49, 61	eqtrd 2656	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopOpen‘𝐾))
63	29, 42, 62	3jca 1242	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 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 {cpr 4179 ⟨cop 4183 ∪ cuni 4436 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 1c1 9937 3c3 11071 9c9 11077 cdc 11493 ndxcnx 15854 sSet csts 15855 Basecbs 15857 TopSetcts 15947 UnifSetcunif 15951 ↾_t crest 16081 TopOpenctopn 16082 UnifOncust 22003 unifTopcutop 22034 toUnifSpctus 22059

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-tset 15960 df-unif 15965 df-rest 16083 df-topn 16084 df-ust 22004 df-utop 22035 df-tus 22062

This theorem is referenced by: tusbas 22072 tusunif 22073 tustopn 22075 tususp 22076

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