cvmlift3lem6 - Mathbox for Mario Carneiro

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Theorem cvmlift3lem6 32859

Description: Lemma for cvmlift3 32863. (Contributed by Mario Carneiro, 9-Jul-2015.)

**Hypotheses**
Ref	Expression
cvmlift3.b	⊢ 𝐵 = ∪ 𝐶
cvmlift3.y	⊢ 𝑌 = ∪ 𝐾
cvmlift3.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmlift3.k	⊢ (𝜑 → 𝐾 ∈ SConn)
cvmlift3.l	⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)
cvmlift3.o	⊢ (𝜑 → 𝑂 ∈ 𝑌)
cvmlift3.g	⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
cvmlift3.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
cvmlift3.e	⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))
cvmlift3.h	⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
cvmlift3lem7.s	⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (^◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾_t 𝑐)Homeo(𝐽 ↾_t 𝑘))))})
cvmlift3lem7.1	⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)
cvmlift3lem7.2	⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴))
cvmlift3lem7.3	⊢ (𝜑 → 𝑀 ⊆ (^◡𝐺 “ 𝐴))
cvmlift3lem7.w	⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝐻‘𝑋) ∈ 𝑏)
cvmlift3lem6.x	⊢ (𝜑 → 𝑋 ∈ 𝑀)
cvmlift3lem6.z	⊢ (𝜑 → 𝑍 ∈ 𝑀)
cvmlift3lem6.q	⊢ (𝜑 → 𝑄 ∈ (II Cn 𝐾))
cvmlift3lem6.r	⊢ 𝑅 = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑄) ∧ (𝑔‘0) = 𝑃))
cvmlift3lem6.1	⊢ (𝜑 → ((𝑄‘0) = 𝑂 ∧ (𝑄‘1) = 𝑋 ∧ (𝑅‘1) = (𝐻‘𝑋)))
cvmlift3lem6.n	⊢ (𝜑 → 𝑁 ∈ (II Cn (𝐾 ↾_t 𝑀)))
cvmlift3lem6.2	⊢ (𝜑 → ((𝑁‘0) = 𝑋 ∧ (𝑁‘1) = 𝑍))
cvmlift3lem6.i	⊢ 𝐼 = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = (𝐻‘𝑋)))

**Assertion**
Ref	Expression
cvmlift3lem6	⊢ (𝜑 → (𝐻‘𝑍) ∈ 𝑊)

Distinct variable groups: 𝑏,𝑐,𝑑,𝑓,𝑘,𝑠,𝑧,𝐴 𝑓,𝑔,𝐼,𝑧 𝑔,𝑏,𝑥,𝐽,𝑐,𝑑,𝑓,𝑘,𝑠 𝐹,𝑏,𝑐,𝑑,𝑓,𝑔,𝑘,𝑠 𝑥,𝑧,𝐹 𝑓,𝑀,𝑔,𝑥 𝑓,𝑁,𝑔 𝐻,𝑏,𝑐,𝑑,𝑓,𝑔,𝑥,𝑧 𝑄,𝑓,𝑔 𝑆,𝑏,𝑓,𝑥 𝐵,𝑏,𝑑,𝑓,𝑔,𝑥,𝑧 𝑅,𝑔 𝑋,𝑏,𝑐,𝑑,𝑓,𝑔,𝑥,𝑧 𝐺,𝑏,𝑐,𝑑,𝑓,𝑔,𝑘,𝑥,𝑧 𝑇,𝑏,𝑐,𝑑,𝑠 𝑓,𝑍,𝑔,𝑥,𝑧 𝐶,𝑏,𝑐,𝑑,𝑓,𝑔,𝑘,𝑠,𝑥,𝑧 𝜑,𝑓,𝑥 𝐾,𝑏,𝑐,𝑓,𝑔,𝑥,𝑧 𝑃,𝑏,𝑐,𝑑,𝑓,𝑔,𝑥,𝑧 𝑂,𝑏,𝑐,𝑓,𝑔,𝑥,𝑧 𝑓,𝑌,𝑔,𝑥,𝑧 𝑊,𝑐,𝑑,𝑓,𝑥 Allowed substitution hints: 𝜑(𝑧,𝑔,𝑘,𝑠,𝑏,𝑐,𝑑) 𝐴(𝑥,𝑔) 𝐵(𝑘,𝑠,𝑐) 𝑃(𝑘,𝑠) 𝑄(𝑥,𝑧,𝑘,𝑠,𝑏,𝑐,𝑑) 𝑅(𝑥,𝑧,𝑓,𝑘,𝑠,𝑏,𝑐,𝑑) 𝑆(𝑧,𝑔,𝑘,𝑠,𝑐,𝑑) 𝑇(𝑥,𝑧,𝑓,𝑔,𝑘) 𝐺(𝑠) 𝐻(𝑘,𝑠) 𝐼(𝑥,𝑘,𝑠,𝑏,𝑐,𝑑) 𝐽(𝑧) 𝐾(𝑘,𝑠,𝑑) 𝑀(𝑧,𝑘,𝑠,𝑏,𝑐,𝑑) 𝑁(𝑥,𝑧,𝑘,𝑠,𝑏,𝑐,𝑑) 𝑂(𝑘,𝑠,𝑑) 𝑊(𝑧,𝑔,𝑘,𝑠,𝑏) 𝑋(𝑘,𝑠) 𝑌(𝑘,𝑠,𝑏,𝑐,𝑑) 𝑍(𝑘,𝑠,𝑏,𝑐,𝑑)

**Proof of Theorem cvmlift3lem6**
Step	Hyp	Ref	Expression
1		cvmlift3lem6.q	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (II Cn 𝐾))
2		cvmlift3.k	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ SConn)
3		sconntop 32763	. . . . . . . 8 ⊢ (𝐾 ∈ SConn → 𝐾 ∈ Top)
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
5		cnrest2r 22040	. . . . . . 7 ⊢ (𝐾 ∈ Top → (II Cn (𝐾 ↾_t 𝑀)) ⊆ (II Cn 𝐾))
6	4, 5	syl 17	. . . . . 6 ⊢ (𝜑 → (II Cn (𝐾 ↾_t 𝑀)) ⊆ (II Cn 𝐾))
7		cvmlift3lem6.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (II Cn (𝐾 ↾_t 𝑀)))
8	6, 7	sseldd 3878	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (II Cn 𝐾))
9		cvmlift3lem6.1	. . . . . . 7 ⊢ (𝜑 → ((𝑄‘0) = 𝑂 ∧ (𝑄‘1) = 𝑋 ∧ (𝑅‘1) = (𝐻‘𝑋)))
10	9	simp2d 1144	. . . . . 6 ⊢ (𝜑 → (𝑄‘1) = 𝑋)
11		cvmlift3lem6.2	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘0) = 𝑋 ∧ (𝑁‘1) = 𝑍))
12	11	simpld 498	. . . . . 6 ⊢ (𝜑 → (𝑁‘0) = 𝑋)
13	10, 12	eqtr4d 2776	. . . . 5 ⊢ (𝜑 → (𝑄‘1) = (𝑁‘0))
14	1, 8, 13	pcocn 23771	. . . 4 ⊢ (𝜑 → (𝑄(*_𝑝‘𝐾)𝑁) ∈ (II Cn 𝐾))
15	1, 8	pco0 23768	. . . . 5 ⊢ (𝜑 → ((𝑄(*_𝑝‘𝐾)𝑁)‘0) = (𝑄‘0))
16	9	simp1d 1143	. . . . 5 ⊢ (𝜑 → (𝑄‘0) = 𝑂)
17	15, 16	eqtrd 2773	. . . 4 ⊢ (𝜑 → ((𝑄(*_𝑝‘𝐾)𝑁)‘0) = 𝑂)
18	1, 8	pco1 23769	. . . . 5 ⊢ (𝜑 → ((𝑄(*_𝑝‘𝐾)𝑁)‘1) = (𝑁‘1))
19	11	simprd 499	. . . . 5 ⊢ (𝜑 → (𝑁‘1) = 𝑍)
20	18, 19	eqtrd 2773	. . . 4 ⊢ (𝜑 → ((𝑄(*_𝑝‘𝐾)𝑁)‘1) = 𝑍)
21		cvmlift3.b	. . . . . . . . . . 11 ⊢ 𝐵 = ∪ 𝐶
22		cvmlift3lem6.r	. . . . . . . . . . 11 ⊢ 𝑅 = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑄) ∧ (𝑔‘0) = 𝑃))
23		cvmlift3.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
24		cvmlift3.g	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
25		cnco 22019	. . . . . . . . . . . 12 ⊢ ((𝑄 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺 ∘ 𝑄) ∈ (II Cn 𝐽))
26	1, 24, 25	syl2anc 587	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 ∘ 𝑄) ∈ (II Cn 𝐽))
27		cvmlift3.p	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
28	16	fveq2d 6680	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺‘(𝑄‘0)) = (𝐺‘𝑂))
29		iiuni 23635	. . . . . . . . . . . . . . 15 ⊢ (0[,]1) = ∪ II
30		cvmlift3.y	. . . . . . . . . . . . . . 15 ⊢ 𝑌 = ∪ 𝐾
31	29, 30	cnf 21999	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ (II Cn 𝐾) → 𝑄:(0[,]1)⟶𝑌)
32	1, 31	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄:(0[,]1)⟶𝑌)
33		0elunit 12945	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0[,]1)
34		fvco3 6769	. . . . . . . . . . . . 13 ⊢ ((𝑄:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ 𝑄)‘0) = (𝐺‘(𝑄‘0)))
35	32, 33, 34	sylancl 589	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐺 ∘ 𝑄)‘0) = (𝐺‘(𝑄‘0)))
36		cvmlift3.e	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))
37	28, 35, 36	3eqtr4rd 2784	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘𝑃) = ((𝐺 ∘ 𝑄)‘0))
38	21, 22, 23, 26, 27, 37	cvmliftiota 32836	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝑅) = (𝐺 ∘ 𝑄) ∧ (𝑅‘0) = 𝑃))
39	38	simp2d 1144	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∘ 𝑅) = (𝐺 ∘ 𝑄))
40		cvmlift3lem6.i	. . . . . . . . . . 11 ⊢ 𝐼 = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑁) ∧ (𝑔‘0) = (𝐻‘𝑋)))
41		cnco 22019	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺 ∘ 𝑁) ∈ (II Cn 𝐽))
42	8, 24, 41	syl2anc 587	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 ∘ 𝑁) ∈ (II Cn 𝐽))
43		cvmlift3.l	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)
44		cvmlift3.o	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑂 ∈ 𝑌)
45		cvmlift3.h	. . . . . . . . . . . . 13 ⊢ 𝐻 = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
46	21, 30, 23, 2, 43, 44, 24, 27, 36, 45	cvmlift3lem3 32856	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻:𝑌⟶𝐵)
47		cvmlift3lem7.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ⊆ (^◡𝐺 “ 𝐴))
48		cnvimass 5923	. . . . . . . . . . . . . . 15 ⊢ (^◡𝐺 “ 𝐴) ⊆ dom 𝐺
49		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝐽 = ∪ 𝐽
50	30, 49	cnf 21999	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌⟶∪ 𝐽)
51	24, 50	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺:𝑌⟶∪ 𝐽)
52	48, 51	fssdm 6524	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (^◡𝐺 “ 𝐴) ⊆ 𝑌)
53	47, 52	sstrd 3887	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ⊆ 𝑌)
54		cvmlift3lem6.x	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ 𝑀)
55	53, 54	sseldd 3878	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝑌)
56	46, 55	ffvelrnd 6864	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐻‘𝑋) ∈ 𝐵)
57	12	fveq2d 6680	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺‘(𝑁‘0)) = (𝐺‘𝑋))
58	29, 30	cnf 21999	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (II Cn 𝐾) → 𝑁:(0[,]1)⟶𝑌)
59	8, 58	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁:(0[,]1)⟶𝑌)
60		fvco3 6769	. . . . . . . . . . . . 13 ⊢ ((𝑁:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ 𝑁)‘0) = (𝐺‘(𝑁‘0)))
61	59, 33, 60	sylancl 589	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐺 ∘ 𝑁)‘0) = (𝐺‘(𝑁‘0)))
62		fvco3 6769	. . . . . . . . . . . . . 14 ⊢ ((𝐻:𝑌⟶𝐵 ∧ 𝑋 ∈ 𝑌) → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐹‘(𝐻‘𝑋)))
63	46, 55, 62	syl2anc 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐹‘(𝐻‘𝑋)))
64	21, 30, 23, 2, 43, 44, 24, 27, 36, 45	cvmlift3lem5 32858	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ∘ 𝐻) = 𝐺)
65	64	fveq1d 6678	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐺‘𝑋))
66	63, 65	eqtr3d 2775	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘(𝐻‘𝑋)) = (𝐺‘𝑋))
67	57, 61, 66	3eqtr4rd 2784	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘(𝐻‘𝑋)) = ((𝐺 ∘ 𝑁)‘0))
68	21, 40, 23, 42, 56, 67	cvmliftiota 32836	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐼) = (𝐺 ∘ 𝑁) ∧ (𝐼‘0) = (𝐻‘𝑋)))
69	68	simp2d 1144	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∘ 𝐼) = (𝐺 ∘ 𝑁))
70	39, 69	oveq12d 7190	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 ∘ 𝑅)(_𝑝‘𝐽)(𝐹 ∘ 𝐼)) = ((𝐺 ∘ 𝑄)(_𝑝‘𝐽)(𝐺 ∘ 𝑁)))
71	38	simp1d 1143	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ (II Cn 𝐶))
72	68	simp1d 1143	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐶))
73	9	simp3d 1145	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅‘1) = (𝐻‘𝑋))
74	68	simp3d 1145	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘0) = (𝐻‘𝑋))
75	73, 74	eqtr4d 2776	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘1) = (𝐼‘0))
76		cvmcn 32797	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
77	23, 76	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Cn 𝐽))
78	71, 72, 75, 77	copco 23772	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (𝑅(_𝑝‘𝐶)𝐼)) = ((𝐹 ∘ 𝑅)(_𝑝‘𝐽)(𝐹 ∘ 𝐼)))
79	1, 8, 13, 24	copco 23772	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) = ((𝐺 ∘ 𝑄)(_𝑝‘𝐽)(𝐺 ∘ 𝑁)))
80	70, 78, 79	3eqtr4d 2783	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ (𝑅(_𝑝‘𝐶)𝐼)) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)))
81	71, 72	pco0 23768	. . . . . . . 8 ⊢ (𝜑 → ((𝑅(*_𝑝‘𝐶)𝐼)‘0) = (𝑅‘0))
82	38	simp3d 1145	. . . . . . . 8 ⊢ (𝜑 → (𝑅‘0) = 𝑃)
83	81, 82	eqtrd 2773	. . . . . . 7 ⊢ (𝜑 → ((𝑅(*_𝑝‘𝐶)𝐼)‘0) = 𝑃)
84	71, 72, 75	pcocn 23771	. . . . . . . 8 ⊢ (𝜑 → (𝑅(*_𝑝‘𝐶)𝐼) ∈ (II Cn 𝐶))
85		cnco 22019	. . . . . . . . . 10 ⊢ (((𝑄(_𝑝‘𝐾)𝑁) ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) ∈ (II Cn 𝐽))
86	14, 24, 85	syl2anc 587	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ∘ (𝑄(*_𝑝‘𝐾)𝑁)) ∈ (II Cn 𝐽))
87	17	fveq2d 6680	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘((𝑄(*_𝑝‘𝐾)𝑁)‘0)) = (𝐺‘𝑂))
88	29, 30	cnf 21999	. . . . . . . . . . . 12 ⊢ ((𝑄(_𝑝‘𝐾)𝑁) ∈ (II Cn 𝐾) → (𝑄(_𝑝‘𝐾)𝑁):(0[,]1)⟶𝑌)
89	14, 88	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄(*_𝑝‘𝐾)𝑁):(0[,]1)⟶𝑌)
90		fvco3 6769	. . . . . . . . . . 11 ⊢ (((𝑄(_𝑝‘𝐾)𝑁):(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁))‘0) = (𝐺‘((𝑄(*_𝑝‘𝐾)𝑁)‘0)))
91	89, 33, 90	sylancl 589	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁))‘0) = (𝐺‘((𝑄(_𝑝‘𝐾)𝑁)‘0)))
92	87, 91, 36	3eqtr4rd 2784	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑃) = ((𝐺 ∘ (𝑄(*_𝑝‘𝐾)𝑁))‘0))
93	21	cvmlift 32834	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) ∈ (II Cn 𝐽)) ∧ (𝑃 ∈ 𝐵 ∧ (𝐹‘𝑃) = ((𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁))‘0))) → ∃!𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(*_𝑝‘𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))
94	23, 86, 27, 92, 93	syl22anc 838	. . . . . . . 8 ⊢ (𝜑 → ∃!𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(*_𝑝‘𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))
95		coeq2 5701	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑅(_𝑝‘𝐶)𝐼) → (𝐹 ∘ 𝑔) = (𝐹 ∘ (𝑅(_𝑝‘𝐶)𝐼)))
96	95	eqeq1d 2740	. . . . . . . . . 10 ⊢ (𝑔 = (𝑅(_𝑝‘𝐶)𝐼) → ((𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) ↔ (𝐹 ∘ (𝑅(_𝑝‘𝐶)𝐼)) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁))))
97		fveq1 6675	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑅(_𝑝‘𝐶)𝐼) → (𝑔‘0) = ((𝑅(_𝑝‘𝐶)𝐼)‘0))
98	97	eqeq1d 2740	. . . . . . . . . 10 ⊢ (𝑔 = (𝑅(_𝑝‘𝐶)𝐼) → ((𝑔‘0) = 𝑃 ↔ ((𝑅(_𝑝‘𝐶)𝐼)‘0) = 𝑃))
99	96, 98	anbi12d 634	. . . . . . . . 9 ⊢ (𝑔 = (𝑅(_𝑝‘𝐶)𝐼) → (((𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ (𝑅(_𝑝‘𝐶)𝐼)) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) ∧ ((𝑅(*_𝑝‘𝐶)𝐼)‘0) = 𝑃)))
100	99	riota2 7155	. . . . . . . 8 ⊢ (((𝑅(_𝑝‘𝐶)𝐼) ∈ (II Cn 𝐶) ∧ ∃!𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃)) → (((𝐹 ∘ (𝑅(_𝑝‘𝐶)𝐼)) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) ∧ ((𝑅(_𝑝‘𝐶)𝐼)‘0) = 𝑃) ↔ (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃)) = (𝑅(*_𝑝‘𝐶)𝐼)))
101	84, 94, 100	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → (((𝐹 ∘ (𝑅(_𝑝‘𝐶)𝐼)) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) ∧ ((𝑅(_𝑝‘𝐶)𝐼)‘0) = 𝑃) ↔ (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃)) = (𝑅(*_𝑝‘𝐶)𝐼)))
102	80, 83, 101	mpbi2and 712	. . . . . 6 ⊢ (𝜑 → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃)) = (𝑅(_𝑝‘𝐶)𝐼))
103	102	fveq1d 6678	. . . . 5 ⊢ (𝜑 → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = ((𝑅(_𝑝‘𝐶)𝐼)‘1))
104	71, 72	pco1 23769	. . . . 5 ⊢ (𝜑 → ((𝑅(*_𝑝‘𝐶)𝐼)‘1) = (𝐼‘1))
105	103, 104	eqtrd 2773	. . . 4 ⊢ (𝜑 → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(*_𝑝‘𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1))
106		fveq1 6675	. . . . . . 7 ⊢ (𝑓 = (𝑄(_𝑝‘𝐾)𝑁) → (𝑓‘0) = ((𝑄(_𝑝‘𝐾)𝑁)‘0))
107	106	eqeq1d 2740	. . . . . 6 ⊢ (𝑓 = (𝑄(_𝑝‘𝐾)𝑁) → ((𝑓‘0) = 𝑂 ↔ ((𝑄(_𝑝‘𝐾)𝑁)‘0) = 𝑂))
108		fveq1 6675	. . . . . . 7 ⊢ (𝑓 = (𝑄(_𝑝‘𝐾)𝑁) → (𝑓‘1) = ((𝑄(_𝑝‘𝐾)𝑁)‘1))
109	108	eqeq1d 2740	. . . . . 6 ⊢ (𝑓 = (𝑄(_𝑝‘𝐾)𝑁) → ((𝑓‘1) = 𝑍 ↔ ((𝑄(_𝑝‘𝐾)𝑁)‘1) = 𝑍))
110		coeq2 5701	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑄(_𝑝‘𝐾)𝑁) → (𝐺 ∘ 𝑓) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)))
111	110	eqeq2d 2749	. . . . . . . . . 10 ⊢ (𝑓 = (𝑄(_𝑝‘𝐾)𝑁) → ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁))))
112	111	anbi1d 633	. . . . . . . . 9 ⊢ (𝑓 = (𝑄(_𝑝‘𝐾)𝑁) → (((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃)))
113	112	riotabidv 7131	. . . . . . . 8 ⊢ (𝑓 = (𝑄(_𝑝‘𝐾)𝑁) → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃)))
114	113	fveq1d 6678	. . . . . . 7 ⊢ (𝑓 = (𝑄(_𝑝‘𝐾)𝑁) → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1))
115	114	eqeq1d 2740	. . . . . 6 ⊢ (𝑓 = (𝑄(_𝑝‘𝐾)𝑁) → (((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1) ↔ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1)))
116	107, 109, 115	3anbi123d 1437	. . . . 5 ⊢ (𝑓 = (𝑄(_𝑝‘𝐾)𝑁) → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑍 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1)) ↔ (((𝑄(_𝑝‘𝐾)𝑁)‘0) = 𝑂 ∧ ((𝑄(_𝑝‘𝐾)𝑁)‘1) = 𝑍 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1))))
117	116	rspcev 3526	. . . 4 ⊢ (((𝑄(_𝑝‘𝐾)𝑁) ∈ (II Cn 𝐾) ∧ (((𝑄(_𝑝‘𝐾)𝑁)‘0) = 𝑂 ∧ ((𝑄(_𝑝‘𝐾)𝑁)‘1) = 𝑍 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1))) → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑍 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1)))
118	14, 17, 20, 105, 117	syl13anc 1373	. . 3 ⊢ (𝜑 → ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑍 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1)))
119		cvmlift3lem6.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑀)
120	53, 119	sseldd 3878	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑌)
121	21, 30, 23, 2, 43, 44, 24, 27, 36, 45	cvmlift3lem4 32857	. . . 4 ⊢ ((𝜑 ∧ 𝑍 ∈ 𝑌) → ((𝐻‘𝑍) = (𝐼‘1) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑍 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1))))
122	120, 121	mpdan 687	. . 3 ⊢ (𝜑 → ((𝐻‘𝑍) = (𝐼‘1) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑍 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1))))
123	118, 122	mpbird 260	. 2 ⊢ (𝜑 → (𝐻‘𝑍) = (𝐼‘1))
124		iiconn 23641	. . . . 5 ⊢ II ∈ Conn
125	124	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ Conn)
126		cvmtop1 32795	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
127	23, 126	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Top)
128	21	toptopon 21670	. . . . . . 7 ⊢ (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
129	127, 128	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (TopOn‘𝐵))
130	69	rneqd 5781	. . . . . . . . 9 ⊢ (𝜑 → ran (𝐹 ∘ 𝐼) = ran (𝐺 ∘ 𝑁))
131		rnco2 6086	. . . . . . . . 9 ⊢ ran (𝐹 ∘ 𝐼) = (𝐹 “ ran 𝐼)
132		rnco2 6086	. . . . . . . . 9 ⊢ ran (𝐺 ∘ 𝑁) = (𝐺 “ ran 𝑁)
133	130, 131, 132	3eqtr3g 2796	. . . . . . . 8 ⊢ (𝜑 → (𝐹 “ ran 𝐼) = (𝐺 “ ran 𝑁))
134		iitopon 23633	. . . . . . . . . . . . 13 ⊢ II ∈ (TopOn‘(0[,]1))
135	134	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
136	30	toptopon 21670	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
137	4, 136	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
138		resttopon 21914	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑀 ⊆ 𝑌) → (𝐾 ↾_t 𝑀) ∈ (TopOn‘𝑀))
139	137, 53, 138	syl2anc 587	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾 ↾_t 𝑀) ∈ (TopOn‘𝑀))
140		cnf2 22002	. . . . . . . . . . . 12 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ (𝐾 ↾_t 𝑀) ∈ (TopOn‘𝑀) ∧ 𝑁 ∈ (II Cn (𝐾 ↾_t 𝑀))) → 𝑁:(0[,]1)⟶𝑀)
141	135, 139, 7, 140	syl3anc 1372	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁:(0[,]1)⟶𝑀)
142	141	frnd 6512	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑁 ⊆ 𝑀)
143	142, 47	sstrd 3887	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑁 ⊆ (^◡𝐺 “ 𝐴))
144	51	ffund 6508	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝐺)
145	143, 48	sstrdi 3889	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑁 ⊆ dom 𝐺)
146		funimass3 6833	. . . . . . . . . 10 ⊢ ((Fun 𝐺 ∧ ran 𝑁 ⊆ dom 𝐺) → ((𝐺 “ ran 𝑁) ⊆ 𝐴 ↔ ran 𝑁 ⊆ (^◡𝐺 “ 𝐴)))
147	144, 145, 146	syl2anc 587	. . . . . . . . 9 ⊢ (𝜑 → ((𝐺 “ ran 𝑁) ⊆ 𝐴 ↔ ran 𝑁 ⊆ (^◡𝐺 “ 𝐴)))
148	143, 147	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → (𝐺 “ ran 𝑁) ⊆ 𝐴)
149	133, 148	eqsstrd 3915	. . . . . . 7 ⊢ (𝜑 → (𝐹 “ ran 𝐼) ⊆ 𝐴)
150	21, 49	cnf 21999	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶∪ 𝐽)
151	77, 150	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐵⟶∪ 𝐽)
152	151	ffund 6508	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
153	29, 21	cnf 21999	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (II Cn 𝐶) → 𝐼:(0[,]1)⟶𝐵)
154	72, 153	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼:(0[,]1)⟶𝐵)
155	154	frnd 6512	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐼 ⊆ 𝐵)
156	151	fdmd 6515	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝐵)
157	155, 156	sseqtrrd 3918	. . . . . . . 8 ⊢ (𝜑 → ran 𝐼 ⊆ dom 𝐹)
158		funimass3 6833	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ ran 𝐼 ⊆ dom 𝐹) → ((𝐹 “ ran 𝐼) ⊆ 𝐴 ↔ ran 𝐼 ⊆ (^◡𝐹 “ 𝐴)))
159	152, 157, 158	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → ((𝐹 “ ran 𝐼) ⊆ 𝐴 ↔ ran 𝐼 ⊆ (^◡𝐹 “ 𝐴)))
160	149, 159	mpbid 235	. . . . . 6 ⊢ (𝜑 → ran 𝐼 ⊆ (^◡𝐹 “ 𝐴))
161		cnvimass 5923	. . . . . . 7 ⊢ (^◡𝐹 “ 𝐴) ⊆ dom 𝐹
162	161, 151	fssdm 6524	. . . . . 6 ⊢ (𝜑 → (^◡𝐹 “ 𝐴) ⊆ 𝐵)
163		cnrest2 22039	. . . . . 6 ⊢ ((𝐶 ∈ (TopOn‘𝐵) ∧ ran 𝐼 ⊆ (^◡𝐹 “ 𝐴) ∧ (^◡𝐹 “ 𝐴) ⊆ 𝐵) → (𝐼 ∈ (II Cn 𝐶) ↔ 𝐼 ∈ (II Cn (𝐶 ↾_t (^◡𝐹 “ 𝐴)))))
164	129, 160, 162, 163	syl3anc 1372	. . . . 5 ⊢ (𝜑 → (𝐼 ∈ (II Cn 𝐶) ↔ 𝐼 ∈ (II Cn (𝐶 ↾_t (^◡𝐹 “ 𝐴)))))
165	72, 164	mpbid 235	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (II Cn (𝐶 ↾_t (^◡𝐹 “ 𝐴))))
166		cvmlift3lem7.2	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ (𝑆‘𝐴))
167		cvmlift3lem7.s	. . . . . . . 8 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (^◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾_t 𝑐)Homeo(𝐽 ↾_t 𝑘))))})
168	167	cvmsss 32802	. . . . . . 7 ⊢ (𝑇 ∈ (𝑆‘𝐴) → 𝑇 ⊆ 𝐶)
169	166, 168	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑇 ⊆ 𝐶)
170		cvmlift3lem7.1	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘𝑋) ∈ 𝐴)
171	66, 170	eqeltrd 2833	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝐻‘𝑋)) ∈ 𝐴)
172		cvmlift3lem7.w	. . . . . . . . 9 ⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝐻‘𝑋) ∈ 𝑏)
173	167, 21, 172	cvmsiota 32812	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ (𝑇 ∈ (𝑆‘𝐴) ∧ (𝐻‘𝑋) ∈ 𝐵 ∧ (𝐹‘(𝐻‘𝑋)) ∈ 𝐴)) → (𝑊 ∈ 𝑇 ∧ (𝐻‘𝑋) ∈ 𝑊))
174	23, 166, 56, 171, 173	syl13anc 1373	. . . . . . 7 ⊢ (𝜑 → (𝑊 ∈ 𝑇 ∧ (𝐻‘𝑋) ∈ 𝑊))
175	174	simpld 498	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝑇)
176	169, 175	sseldd 3878	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐶)
177		elssuni 4828	. . . . . . 7 ⊢ (𝑊 ∈ 𝑇 → 𝑊 ⊆ ∪ 𝑇)
178	175, 177	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ⊆ ∪ 𝑇)
179	167	cvmsuni 32804	. . . . . . 7 ⊢ (𝑇 ∈ (𝑆‘𝐴) → ∪ 𝑇 = (^◡𝐹 “ 𝐴))
180	166, 179	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝑇 = (^◡𝐹 “ 𝐴))
181	178, 180	sseqtrd 3917	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ (^◡𝐹 “ 𝐴))
182	167	cvmsrcl 32799	. . . . . . . 8 ⊢ (𝑇 ∈ (𝑆‘𝐴) → 𝐴 ∈ 𝐽)
183	166, 182	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐽)
184		cnima 22018	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 Cn 𝐽) ∧ 𝐴 ∈ 𝐽) → (^◡𝐹 “ 𝐴) ∈ 𝐶)
185	77, 183, 184	syl2anc 587	. . . . . 6 ⊢ (𝜑 → (^◡𝐹 “ 𝐴) ∈ 𝐶)
186		restopn2 21930	. . . . . 6 ⊢ ((𝐶 ∈ Top ∧ (^◡𝐹 “ 𝐴) ∈ 𝐶) → (𝑊 ∈ (𝐶 ↾_t (^◡𝐹 “ 𝐴)) ↔ (𝑊 ∈ 𝐶 ∧ 𝑊 ⊆ (^◡𝐹 “ 𝐴))))
187	127, 185, 186	syl2anc 587	. . . . 5 ⊢ (𝜑 → (𝑊 ∈ (𝐶 ↾_t (^◡𝐹 “ 𝐴)) ↔ (𝑊 ∈ 𝐶 ∧ 𝑊 ⊆ (^◡𝐹 “ 𝐴))))
188	176, 181, 187	mpbir2and 713	. . . 4 ⊢ (𝜑 → 𝑊 ∈ (𝐶 ↾_t (^◡𝐹 “ 𝐴)))
189	167	cvmscld 32808	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝐴) ∧ 𝑊 ∈ 𝑇) → 𝑊 ∈ (Clsd‘(𝐶 ↾_t (^◡𝐹 “ 𝐴))))
190	23, 166, 175, 189	syl3anc 1372	. . . 4 ⊢ (𝜑 → 𝑊 ∈ (Clsd‘(𝐶 ↾_t (^◡𝐹 “ 𝐴))))
191	33	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ (0[,]1))
192	174	simprd 499	. . . . 5 ⊢ (𝜑 → (𝐻‘𝑋) ∈ 𝑊)
193	74, 192	eqeltrd 2833	. . . 4 ⊢ (𝜑 → (𝐼‘0) ∈ 𝑊)
194	29, 125, 165, 188, 190, 191, 193	conncn 22179	. . 3 ⊢ (𝜑 → 𝐼:(0[,]1)⟶𝑊)
195		1elunit 12946	. . 3 ⊢ 1 ∈ (0[,]1)
196		ffvelrn 6861	. . 3 ⊢ ((𝐼:(0[,]1)⟶𝑊 ∧ 1 ∈ (0[,]1)) → (𝐼‘1) ∈ 𝑊)
197	194, 195, 196	sylancl 589	. 2 ⊢ (𝜑 → (𝐼‘1) ∈ 𝑊)
198	123, 197	eqeltrd 2833	1 ⊢ (𝜑 → (𝐻‘𝑍) ∈ 𝑊)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3053 ∃wrex 3054 ∃!wreu 3055 {crab 3057 ∖ cdif 3840 ∩ cin 3842 ⊆ wss 3843 ∅c0 4211 𝒫 cpw 4488 {csn 4516 ∪ cuni 4796 ↦ cmpt 5110 ^◡ccnv 5524 dom cdm 5525 ran crn 5526 ↾ cres 5527 “ cima 5528 ∘ ccom 5529 Fun wfun 6333 ⟶wf 6335 ‘cfv 6339 ℩crio 7128 (class class class)co 7172 0cc0 10617 1c1 10618 [,]cicc 12826 ↾_t crest 16799 Topctop 21646 TopOnctopon 21663 Clsdccld 21769 Cn ccn 21977 Conncconn 22164 𝑛-Locally cnlly 22218 Homeochmeo 22506 IIcii 23629 *_𝑝cpco 23754 PConncpconn 32754 SConncsconn 32755 CovMap ccvm 32790

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 7481 ax-inf2 9179 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 ax-pre-sup 10695 ax-addf 10696 ax-mulf 10697

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-rmo 3061 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-int 4837 df-iun 4883 df-iin 4884 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-se 5484 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-isom 6348 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-of 7427 df-om 7602 df-1st 7716 df-2nd 7717 df-supp 7859 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-2o 8134 df-er 8322 df-ec 8324 df-map 8441 df-ixp 8510 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-fsupp 8909 df-fi 8950 df-sup 8981 df-inf 8982 df-oi 9049 df-card 9443 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-div 11378 df-nn 11719 df-2 11781 df-3 11782 df-4 11783 df-5 11784 df-6 11785 df-7 11786 df-8 11787 df-9 11788 df-n0 11979 df-z 12065 df-dec 12182 df-uz 12327 df-q 12433 df-rp 12475 df-xneg 12592 df-xadd 12593 df-xmul 12594 df-ioo 12827 df-ico 12829 df-icc 12830 df-fz 12984 df-fzo 13127 df-fl 13255 df-seq 13463 df-exp 13524 df-hash 13785 df-cj 14550 df-re 14551 df-im 14552 df-sqrt 14686 df-abs 14687 df-clim 14937 df-sum 15138 df-struct 16590 df-ndx 16591 df-slot 16592 df-base 16594 df-sets 16595 df-ress 16596 df-plusg 16683 df-mulr 16684 df-starv 16685 df-sca 16686 df-vsca 16687 df-ip 16688 df-tset 16689 df-ple 16690 df-ds 16692 df-unif 16693 df-hom 16694 df-cco 16695 df-rest 16801 df-topn 16802 df-0g 16820 df-gsum 16821 df-topgen 16822 df-pt 16823 df-prds 16826 df-xrs 16880 df-qtop 16885 df-imas 16886 df-xps 16888 df-mre 16962 df-mrc 16963 df-acs 16965 df-mgm 17970 df-sgrp 18019 df-mnd 18030 df-submnd 18075 df-mulg 18345 df-cntz 18567 df-cmn 19028 df-psmet 20211 df-xmet 20212 df-met 20213 df-bl 20214 df-mopn 20215 df-cnfld 20220 df-top 21647 df-topon 21664 df-topsp 21686 df-bases 21699 df-cld 21772 df-ntr 21773 df-cls 21774 df-nei 21851 df-cn 21980 df-cnp 21981 df-cmp 22140 df-conn 22165 df-lly 22219 df-nlly 22220 df-tx 22315 df-hmeo 22508 df-xms 23075 df-ms 23076 df-tms 23077 df-ii 23631 df-htpy 23724 df-phtpy 23725 df-phtpc 23746 df-pco 23759 df-pconn 32756 df-sconn 32757 df-cvm 32791

This theorem is referenced by: cvmlift3lem7 32860

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