cvmlift3lem6 - Mathbox for Mario Carneiro

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Theorem cvmlift3lem6 35356

Description: Lemma for cvmlift3 35360. (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 35260	. . . . . . . 8 ⊢ (𝐾 ∈ SConn → 𝐾 ∈ Top)
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
5		cnrest2r 23200	. . . . . . 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 3935	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (II Cn 𝐾))
9		cvmlift3lem6.1	. . . . . . 7 ⊢ (𝜑 → ((𝑄‘0) = 𝑂 ∧ (𝑄‘1) = 𝑋 ∧ (𝑅‘1) = (𝐻‘𝑋)))
10	9	simp2d 1143	. . . . . 6 ⊢ (𝜑 → (𝑄‘1) = 𝑋)
11		cvmlift3lem6.2	. . . . . . 7 ⊢ (𝜑 → ((𝑁‘0) = 𝑋 ∧ (𝑁‘1) = 𝑍))
12	11	simpld 494	. . . . . 6 ⊢ (𝜑 → (𝑁‘0) = 𝑋)
13	10, 12	eqtr4d 2769	. . . . 5 ⊢ (𝜑 → (𝑄‘1) = (𝑁‘0))
14	1, 8, 13	pcocn 24942	. . . 4 ⊢ (𝜑 → (𝑄(*_𝑝‘𝐾)𝑁) ∈ (II Cn 𝐾))
15	1, 8	pco0 24939	. . . . 5 ⊢ (𝜑 → ((𝑄(*_𝑝‘𝐾)𝑁)‘0) = (𝑄‘0))
16	9	simp1d 1142	. . . . 5 ⊢ (𝜑 → (𝑄‘0) = 𝑂)
17	15, 16	eqtrd 2766	. . . 4 ⊢ (𝜑 → ((𝑄(*_𝑝‘𝐾)𝑁)‘0) = 𝑂)
18	1, 8	pco1 24940	. . . . 5 ⊢ (𝜑 → ((𝑄(*_𝑝‘𝐾)𝑁)‘1) = (𝑁‘1))
19	11	simprd 495	. . . . 5 ⊢ (𝜑 → (𝑁‘1) = 𝑍)
20	18, 19	eqtrd 2766	. . . 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 23179	. . . . . . . . . . . 12 ⊢ ((𝑄 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺 ∘ 𝑄) ∈ (II Cn 𝐽))
26	1, 24, 25	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 ∘ 𝑄) ∈ (II Cn 𝐽))
27		cvmlift3.p	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
28	16	fveq2d 6826	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺‘(𝑄‘0)) = (𝐺‘𝑂))
29		iiuni 24799	. . . . . . . . . . . . . . 15 ⊢ (0[,]1) = ∪ II
30		cvmlift3.y	. . . . . . . . . . . . . . 15 ⊢ 𝑌 = ∪ 𝐾
31	29, 30	cnf 23159	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ (II Cn 𝐾) → 𝑄:(0[,]1)⟶𝑌)
32	1, 31	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄:(0[,]1)⟶𝑌)
33		0elunit 13366	. . . . . . . . . . . . 13 ⊢ 0 ∈ (0[,]1)
34		fvco3 6921	. . . . . . . . . . . . 13 ⊢ ((𝑄:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ 𝑄)‘0) = (𝐺‘(𝑄‘0)))
35	32, 33, 34	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐺 ∘ 𝑄)‘0) = (𝐺‘(𝑄‘0)))
36		cvmlift3.e	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))
37	28, 35, 36	3eqtr4rd 2777	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘𝑃) = ((𝐺 ∘ 𝑄)‘0))
38	21, 22, 23, 26, 27, 37	cvmliftiota 35333	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝑅) = (𝐺 ∘ 𝑄) ∧ (𝑅‘0) = 𝑃))
39	38	simp2d 1143	. . . . . . . . 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 23179	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (II Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐽)) → (𝐺 ∘ 𝑁) ∈ (II Cn 𝐽))
42	8, 24, 41	syl2anc 584	. . . . . . . . . . 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 35353	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻:𝑌⟶𝐵)
47		cvmlift3lem7.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ⊆ (^◡𝐺 “ 𝐴))
48		cnvimass 6031	. . . . . . . . . . . . . . 15 ⊢ (^◡𝐺 “ 𝐴) ⊆ dom 𝐺
49		eqid 2731	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝐽 = ∪ 𝐽
50	30, 49	cnf 23159	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ (𝐾 Cn 𝐽) → 𝐺:𝑌⟶∪ 𝐽)
51	24, 50	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺:𝑌⟶∪ 𝐽)
52	48, 51	fssdm 6670	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (^◡𝐺 “ 𝐴) ⊆ 𝑌)
53	47, 52	sstrd 3945	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ⊆ 𝑌)
54		cvmlift3lem6.x	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ 𝑀)
55	53, 54	sseldd 3935	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ 𝑌)
56	46, 55	ffvelcdmd 7018	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐻‘𝑋) ∈ 𝐵)
57	12	fveq2d 6826	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺‘(𝑁‘0)) = (𝐺‘𝑋))
58	29, 30	cnf 23159	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (II Cn 𝐾) → 𝑁:(0[,]1)⟶𝑌)
59	8, 58	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁:(0[,]1)⟶𝑌)
60		fvco3 6921	. . . . . . . . . . . . 13 ⊢ ((𝑁:(0[,]1)⟶𝑌 ∧ 0 ∈ (0[,]1)) → ((𝐺 ∘ 𝑁)‘0) = (𝐺‘(𝑁‘0)))
61	59, 33, 60	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐺 ∘ 𝑁)‘0) = (𝐺‘(𝑁‘0)))
62		fvco3 6921	. . . . . . . . . . . . . 14 ⊢ ((𝐻:𝑌⟶𝐵 ∧ 𝑋 ∈ 𝑌) → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐹‘(𝐻‘𝑋)))
63	46, 55, 62	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐹‘(𝐻‘𝑋)))
64	21, 30, 23, 2, 43, 44, 24, 27, 36, 45	cvmlift3lem5 35355	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ∘ 𝐻) = 𝐺)
65	64	fveq1d 6824	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹 ∘ 𝐻)‘𝑋) = (𝐺‘𝑋))
66	63, 65	eqtr3d 2768	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘(𝐻‘𝑋)) = (𝐺‘𝑋))
67	57, 61, 66	3eqtr4rd 2777	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘(𝐻‘𝑋)) = ((𝐺 ∘ 𝑁)‘0))
68	21, 40, 23, 42, 56, 67	cvmliftiota 35333	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼 ∈ (II Cn 𝐶) ∧ (𝐹 ∘ 𝐼) = (𝐺 ∘ 𝑁) ∧ (𝐼‘0) = (𝐻‘𝑋)))
69	68	simp2d 1143	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∘ 𝐼) = (𝐺 ∘ 𝑁))
70	39, 69	oveq12d 7364	. . . . . . . 8 ⊢ (𝜑 → ((𝐹 ∘ 𝑅)(_𝑝‘𝐽)(𝐹 ∘ 𝐼)) = ((𝐺 ∘ 𝑄)(_𝑝‘𝐽)(𝐺 ∘ 𝑁)))
71	38	simp1d 1142	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ (II Cn 𝐶))
72	68	simp1d 1142	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐶))
73	9	simp3d 1144	. . . . . . . . . 10 ⊢ (𝜑 → (𝑅‘1) = (𝐻‘𝑋))
74	68	simp3d 1144	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼‘0) = (𝐻‘𝑋))
75	73, 74	eqtr4d 2769	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘1) = (𝐼‘0))
76		cvmcn 35294	. . . . . . . . . 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 24943	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘ (𝑅(_𝑝‘𝐶)𝐼)) = ((𝐹 ∘ 𝑅)(_𝑝‘𝐽)(𝐹 ∘ 𝐼)))
79	1, 8, 13, 24	copco 24943	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)) = ((𝐺 ∘ 𝑄)(_𝑝‘𝐽)(𝐺 ∘ 𝑁)))
80	70, 78, 79	3eqtr4d 2776	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ (𝑅(_𝑝‘𝐶)𝐼)) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)))
81	71, 72	pco0 24939	. . . . . . . 8 ⊢ (𝜑 → ((𝑅(*_𝑝‘𝐶)𝐼)‘0) = (𝑅‘0))
82	38	simp3d 1144	. . . . . . . 8 ⊢ (𝜑 → (𝑅‘0) = 𝑃)
83	81, 82	eqtrd 2766	. . . . . . 7 ⊢ (𝜑 → ((𝑅(*_𝑝‘𝐶)𝐼)‘0) = 𝑃)
84	71, 72, 75	pcocn 24942	. . . . . . . 8 ⊢ (𝜑 → (𝑅(*_𝑝‘𝐶)𝐼) ∈ (II Cn 𝐶))
85		cnco 23179	. . . . . . . . . 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 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ∘ (𝑄(*_𝑝‘𝐾)𝑁)) ∈ (II Cn 𝐽))
87	17	fveq2d 6826	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘((𝑄(*_𝑝‘𝐾)𝑁)‘0)) = (𝐺‘𝑂))
88	29, 30	cnf 23159	. . . . . . . . . . . 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 6921	. . . . . . . . . . 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 586	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁))‘0) = (𝐺‘((𝑄(_𝑝‘𝐾)𝑁)‘0)))
92	87, 91, 36	3eqtr4rd 2777	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑃) = ((𝐺 ∘ (𝑄(*_𝑝‘𝐾)𝑁))‘0))
93	21	cvmlift 35331	. . . . . . . . 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 5798	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑅(_𝑝‘𝐶)𝐼) → (𝐹 ∘ 𝑔) = (𝐹 ∘ (𝑅(_𝑝‘𝐶)𝐼)))
96	95	eqeq1d 2733	. . . . . . . . . 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 6821	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑅(_𝑝‘𝐶)𝐼) → (𝑔‘0) = ((𝑅(_𝑝‘𝐶)𝐼)‘0))
98	97	eqeq1d 2733	. . . . . . . . . 10 ⊢ (𝑔 = (𝑅(_𝑝‘𝐶)𝐼) → ((𝑔‘0) = 𝑃 ↔ ((𝑅(_𝑝‘𝐶)𝐼)‘0) = 𝑃))
99	96, 98	anbi12d 632	. . . . . . . . 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 7328	. . . . . . . 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 584	. . . . . . 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 6824	. . . . 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 24940	. . . . 5 ⊢ (𝜑 → ((𝑅(*_𝑝‘𝐶)𝐼)‘1) = (𝐼‘1))
105	103, 104	eqtrd 2766	. . . 4 ⊢ (𝜑 → ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(*_𝑝‘𝐾)𝑁)) ∧ (𝑔‘0) = 𝑃))‘1) = (𝐼‘1))
106		fveq1 6821	. . . . . . 7 ⊢ (𝑓 = (𝑄(_𝑝‘𝐾)𝑁) → (𝑓‘0) = ((𝑄(_𝑝‘𝐾)𝑁)‘0))
107	106	eqeq1d 2733	. . . . . 6 ⊢ (𝑓 = (𝑄(_𝑝‘𝐾)𝑁) → ((𝑓‘0) = 𝑂 ↔ ((𝑄(_𝑝‘𝐾)𝑁)‘0) = 𝑂))
108		fveq1 6821	. . . . . . 7 ⊢ (𝑓 = (𝑄(_𝑝‘𝐾)𝑁) → (𝑓‘1) = ((𝑄(_𝑝‘𝐾)𝑁)‘1))
109	108	eqeq1d 2733	. . . . . 6 ⊢ (𝑓 = (𝑄(_𝑝‘𝐾)𝑁) → ((𝑓‘1) = 𝑍 ↔ ((𝑄(_𝑝‘𝐾)𝑁)‘1) = 𝑍))
110		coeq2 5798	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑄(_𝑝‘𝐾)𝑁) → (𝐺 ∘ 𝑓) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁)))
111	110	eqeq2d 2742	. . . . . . . . . 10 ⊢ (𝑓 = (𝑄(_𝑝‘𝐾)𝑁) → ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ (𝑄(_𝑝‘𝐾)𝑁))))
112	111	anbi1d 631	. . . . . . . . 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 7305	. . . . . . . 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 6824	. . . . . . 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 2733	. . . . . 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 1438	. . . . 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 3577	. . . 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 1374	. . 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 3935	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑌)
121	21, 30, 23, 2, 43, 44, 24, 27, 36, 45	cvmlift3lem4 35354	. . . 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 257	. 2 ⊢ (𝜑 → (𝐻‘𝑍) = (𝐼‘1))
124		iiconn 24805	. . . . 5 ⊢ II ∈ Conn
125	124	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ Conn)
126		cvmtop1 35292	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
127	23, 126	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Top)
128	21	toptopon 22830	. . . . . . 7 ⊢ (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
129	127, 128	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (TopOn‘𝐵))
130	69	rneqd 5878	. . . . . . . . 9 ⊢ (𝜑 → ran (𝐹 ∘ 𝐼) = ran (𝐺 ∘ 𝑁))
131		rnco2 6201	. . . . . . . . 9 ⊢ ran (𝐹 ∘ 𝐼) = (𝐹 “ ran 𝐼)
132		rnco2 6201	. . . . . . . . 9 ⊢ ran (𝐺 ∘ 𝑁) = (𝐺 “ ran 𝑁)
133	130, 131, 132	3eqtr3g 2789	. . . . . . . 8 ⊢ (𝜑 → (𝐹 “ ran 𝐼) = (𝐺 “ ran 𝑁))
134		iitopon 24797	. . . . . . . . . . . . 13 ⊢ II ∈ (TopOn‘(0[,]1))
135	134	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
136	30	toptopon 22830	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
137	4, 136	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
138		resttopon 23074	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝑀 ⊆ 𝑌) → (𝐾 ↾_t 𝑀) ∈ (TopOn‘𝑀))
139	137, 53, 138	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐾 ↾_t 𝑀) ∈ (TopOn‘𝑀))
140		cnf2 23162	. . . . . . . . . . . 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 1373	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁:(0[,]1)⟶𝑀)
142	141	frnd 6659	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑁 ⊆ 𝑀)
143	142, 47	sstrd 3945	. . . . . . . . 9 ⊢ (𝜑 → ran 𝑁 ⊆ (^◡𝐺 “ 𝐴))
144	51	ffund 6655	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝐺)
145	143, 48	sstrdi 3947	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑁 ⊆ dom 𝐺)
146		funimass3 6987	. . . . . . . . . 10 ⊢ ((Fun 𝐺 ∧ ran 𝑁 ⊆ dom 𝐺) → ((𝐺 “ ran 𝑁) ⊆ 𝐴 ↔ ran 𝑁 ⊆ (^◡𝐺 “ 𝐴)))
147	144, 145, 146	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((𝐺 “ ran 𝑁) ⊆ 𝐴 ↔ ran 𝑁 ⊆ (^◡𝐺 “ 𝐴)))
148	143, 147	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → (𝐺 “ ran 𝑁) ⊆ 𝐴)
149	133, 148	eqsstrd 3969	. . . . . . 7 ⊢ (𝜑 → (𝐹 “ ran 𝐼) ⊆ 𝐴)
150	21, 49	cnf 23159	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶∪ 𝐽)
151	77, 150	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐵⟶∪ 𝐽)
152	151	ffund 6655	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐹)
153	29, 21	cnf 23159	. . . . . . . . . . 11 ⊢ (𝐼 ∈ (II Cn 𝐶) → 𝐼:(0[,]1)⟶𝐵)
154	72, 153	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼:(0[,]1)⟶𝐵)
155	154	frnd 6659	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐼 ⊆ 𝐵)
156	151	fdmd 6661	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝐵)
157	155, 156	sseqtrrd 3972	. . . . . . . 8 ⊢ (𝜑 → ran 𝐼 ⊆ dom 𝐹)
158		funimass3 6987	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ ran 𝐼 ⊆ dom 𝐹) → ((𝐹 “ ran 𝐼) ⊆ 𝐴 ↔ ran 𝐼 ⊆ (^◡𝐹 “ 𝐴)))
159	152, 157, 158	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((𝐹 “ ran 𝐼) ⊆ 𝐴 ↔ ran 𝐼 ⊆ (^◡𝐹 “ 𝐴)))
160	149, 159	mpbid 232	. . . . . 6 ⊢ (𝜑 → ran 𝐼 ⊆ (^◡𝐹 “ 𝐴))
161		cnvimass 6031	. . . . . . 7 ⊢ (^◡𝐹 “ 𝐴) ⊆ dom 𝐹
162	161, 151	fssdm 6670	. . . . . 6 ⊢ (𝜑 → (^◡𝐹 “ 𝐴) ⊆ 𝐵)
163		cnrest2 23199	. . . . . 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 1373	. . . . 5 ⊢ (𝜑 → (𝐼 ∈ (II Cn 𝐶) ↔ 𝐼 ∈ (II Cn (𝐶 ↾_t (^◡𝐹 “ 𝐴)))))
165	72, 164	mpbid 232	. . . 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 35299	. . . . . . 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 2831	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘(𝐻‘𝑋)) ∈ 𝐴)
172		cvmlift3lem7.w	. . . . . . . . 9 ⊢ 𝑊 = (℩𝑏 ∈ 𝑇 (𝐻‘𝑋) ∈ 𝑏)
173	167, 21, 172	cvmsiota 35309	. . . . . . . 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 1374	. . . . . . 7 ⊢ (𝜑 → (𝑊 ∈ 𝑇 ∧ (𝐻‘𝑋) ∈ 𝑊))
175	174	simpld 494	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝑇)
176	169, 175	sseldd 3935	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐶)
177		elssuni 4889	. . . . . . 7 ⊢ (𝑊 ∈ 𝑇 → 𝑊 ⊆ ∪ 𝑇)
178	175, 177	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ⊆ ∪ 𝑇)
179	167	cvmsuni 35301	. . . . . . 7 ⊢ (𝑇 ∈ (𝑆‘𝐴) → ∪ 𝑇 = (^◡𝐹 “ 𝐴))
180	166, 179	syl 17	. . . . . 6 ⊢ (𝜑 → ∪ 𝑇 = (^◡𝐹 “ 𝐴))
181	178, 180	sseqtrd 3971	. . . . 5 ⊢ (𝜑 → 𝑊 ⊆ (^◡𝐹 “ 𝐴))
182	167	cvmsrcl 35296	. . . . . . . 8 ⊢ (𝑇 ∈ (𝑆‘𝐴) → 𝐴 ∈ 𝐽)
183	166, 182	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐽)
184		cnima 23178	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 Cn 𝐽) ∧ 𝐴 ∈ 𝐽) → (^◡𝐹 “ 𝐴) ∈ 𝐶)
185	77, 183, 184	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (^◡𝐹 “ 𝐴) ∈ 𝐶)
186		restopn2 23090	. . . . . 6 ⊢ ((𝐶 ∈ Top ∧ (^◡𝐹 “ 𝐴) ∈ 𝐶) → (𝑊 ∈ (𝐶 ↾_t (^◡𝐹 “ 𝐴)) ↔ (𝑊 ∈ 𝐶 ∧ 𝑊 ⊆ (^◡𝐹 “ 𝐴))))
187	127, 185, 186	syl2anc 584	. . . . 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 35305	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝐴) ∧ 𝑊 ∈ 𝑇) → 𝑊 ∈ (Clsd‘(𝐶 ↾_t (^◡𝐹 “ 𝐴))))
190	23, 166, 175, 189	syl3anc 1373	. . . 4 ⊢ (𝜑 → 𝑊 ∈ (Clsd‘(𝐶 ↾_t (^◡𝐹 “ 𝐴))))
191	33	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ (0[,]1))
192	174	simprd 495	. . . . 5 ⊢ (𝜑 → (𝐻‘𝑋) ∈ 𝑊)
193	74, 192	eqeltrd 2831	. . . 4 ⊢ (𝜑 → (𝐼‘0) ∈ 𝑊)
194	29, 125, 165, 188, 190, 191, 193	conncn 23339	. . 3 ⊢ (𝜑 → 𝐼:(0[,]1)⟶𝑊)
195		1elunit 13367	. . 3 ⊢ 1 ∈ (0[,]1)
196		ffvelcdm 7014	. . 3 ⊢ ((𝐼:(0[,]1)⟶𝑊 ∧ 1 ∈ (0[,]1)) → (𝐼‘1) ∈ 𝑊)
197	194, 195, 196	sylancl 586	. 2 ⊢ (𝜑 → (𝐼‘1) ∈ 𝑊)
198	123, 197	eqeltrd 2831	1 ⊢ (𝜑 → (𝐻‘𝑍) ∈ 𝑊)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ∃wrex 3056 ∃!wreu 3344 {crab 3395 ∖ cdif 3899 ∩ cin 3901 ⊆ wss 3902 ∅c0 4283 𝒫 cpw 4550 {csn 4576 ∪ cuni 4859 ↦ cmpt 5172 ^◡ccnv 5615 dom cdm 5616 ran crn 5617 ↾ cres 5618 “ cima 5619 ∘ ccom 5620 Fun wfun 6475 ⟶wf 6477 ‘cfv 6481 ℩crio 7302 (class class class)co 7346 0cc0 11003 1c1 11004 [,]cicc 13245 ↾_t crest 17321 Topctop 22806 TopOnctopon 22823 Clsdccld 22929 Cn ccn 23137 Conncconn 23324 𝑛-Locally cnlly 23378 Homeochmeo 23666 IIcii 24793 *_𝑝cpco 24925 PConncpconn 35251 SConncsconn 35252 CovMap ccvm 35287

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-pre-sup 11081 ax-addf 11082

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-ec 8624 df-map 8752 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-7 12190 df-8 12191 df-9 12192 df-n0 12379 df-z 12466 df-dec 12586 df-uz 12730 df-q 12844 df-rp 12888 df-xneg 13008 df-xadd 13009 df-xmul 13010 df-ioo 13246 df-ico 13248 df-icc 13249 df-fz 13405 df-fzo 13552 df-fl 13693 df-seq 13906 df-exp 13966 df-hash 14235 df-cj 15003 df-re 15004 df-im 15005 df-sqrt 15139 df-abs 15140 df-clim 15392 df-sum 15591 df-struct 17055 df-sets 17072 df-slot 17090 df-ndx 17102 df-base 17118 df-ress 17139 df-plusg 17171 df-mulr 17172 df-starv 17173 df-sca 17174 df-vsca 17175 df-ip 17176 df-tset 17177 df-ple 17178 df-ds 17180 df-unif 17181 df-hom 17182 df-cco 17183 df-rest 17323 df-topn 17324 df-0g 17342 df-gsum 17343 df-topgen 17344 df-pt 17345 df-prds 17348 df-xrs 17403 df-qtop 17408 df-imas 17409 df-xps 17411 df-mre 17485 df-mrc 17486 df-acs 17488 df-mgm 18545 df-sgrp 18624 df-mnd 18640 df-submnd 18689 df-mulg 18978 df-cntz 19227 df-cmn 19692 df-psmet 21281 df-xmet 21282 df-met 21283 df-bl 21284 df-mopn 21285 df-cnfld 21290 df-top 22807 df-topon 22824 df-topsp 22846 df-bases 22859 df-cld 22932 df-ntr 22933 df-cls 22934 df-nei 23011 df-cn 23140 df-cnp 23141 df-cmp 23300 df-conn 23325 df-lly 23379 df-nlly 23380 df-tx 23475 df-hmeo 23668 df-xms 24233 df-ms 24234 df-tms 24235 df-ii 24795 df-cncf 24796 df-htpy 24894 df-phtpy 24895 df-phtpc 24916 df-pco 24930 df-pconn 35253 df-sconn 35254 df-cvm 35288

This theorem is referenced by: cvmlift3lem7 35357

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