cvmliftmolem1 - Mathbox for Mario Carneiro

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Theorem cvmliftmolem1 32528

Description: Lemma for cvmliftmo 32531. (Contributed by Mario Carneiro, 10-Mar-2015.)

**Hypotheses**
Ref	Expression
cvmliftmo.b	⊢ 𝐵 = ∪ 𝐶
cvmliftmo.y	⊢ 𝑌 = ∪ 𝐾
cvmliftmo.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmliftmo.k	⊢ (𝜑 → 𝐾 ∈ Conn)
cvmliftmo.l	⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn)
cvmliftmo.o	⊢ (𝜑 → 𝑂 ∈ 𝑌)
cvmliftmoi.m	⊢ (𝜑 → 𝑀 ∈ (𝐾 Cn 𝐶))
cvmliftmoi.n	⊢ (𝜑 → 𝑁 ∈ (𝐾 Cn 𝐶))
cvmliftmoi.g	⊢ (𝜑 → (𝐹 ∘ 𝑀) = (𝐹 ∘ 𝑁))
cvmliftmoi.p	⊢ (𝜑 → (𝑀‘𝑂) = (𝑁‘𝑂))
cvmliftmolem.1	⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (^◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾_t 𝑢)Homeo(𝐽 ↾_t 𝑘))))})
cvmliftmolem.2	⊢ ((𝜑 ∧ 𝜓) → 𝑇 ∈ (𝑆‘𝑈))
cvmliftmolem.3	⊢ ((𝜑 ∧ 𝜓) → 𝑊 ∈ 𝑇)
cvmliftmolem.4	⊢ ((𝜑 ∧ 𝜓) → 𝐼 ⊆ (^◡𝑀 “ 𝑊))
cvmliftmolem.5	⊢ ((𝜑 ∧ 𝜓) → (𝐾 ↾_t 𝐼) ∈ Conn)
cvmliftmolem.6	⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐼)
cvmliftmolem.7	⊢ ((𝜑 ∧ 𝜓) → 𝑄 ∈ 𝐼)
cvmliftmolem.8	⊢ ((𝜑 ∧ 𝜓) → 𝑅 ∈ 𝐼)
cvmliftmolem.9	⊢ ((𝜑 ∧ 𝜓) → (𝐹‘(𝑀‘𝑋)) ∈ 𝑈)

**Assertion**
Ref	Expression
cvmliftmolem1	⊢ ((𝜑 ∧ 𝜓) → (𝑄 ∈ dom (𝑀 ∩ 𝑁) → 𝑅 ∈ dom (𝑀 ∩ 𝑁)))

Distinct variable groups: 𝑘,𝑠,𝑢,𝑣,𝐶 𝑘,𝐽,𝑠,𝑢,𝑣 𝑣,𝐵 𝐾,𝑠 𝑘,𝑀,𝑠,𝑢,𝑣 𝑁,𝑠 𝜑,𝑠 𝑘,𝐹,𝑠,𝑢,𝑣 𝑆,𝑠 𝑈,𝑘,𝑠,𝑢,𝑣 𝑇,𝑠,𝑢,𝑣 𝑢,𝑊,𝑣 𝑌,𝑠 Allowed substitution hints: 𝜑(𝑣,𝑢,𝑘) 𝜓(𝑣,𝑢,𝑘,𝑠) 𝐵(𝑢,𝑘,𝑠) 𝑄(𝑣,𝑢,𝑘,𝑠) 𝑅(𝑣,𝑢,𝑘,𝑠) 𝑆(𝑣,𝑢,𝑘) 𝑇(𝑘) 𝐼(𝑣,𝑢,𝑘,𝑠) 𝐾(𝑣,𝑢,𝑘) 𝑁(𝑣,𝑢,𝑘) 𝑂(𝑣,𝑢,𝑘,𝑠) 𝑊(𝑘,𝑠) 𝑋(𝑣,𝑢,𝑘,𝑠) 𝑌(𝑣,𝑢,𝑘)

Proof of Theorem cvmliftmolem1 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmliftmoi.g	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∘ 𝑀) = (𝐹 ∘ 𝑁))
2	1	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝐹 ∘ 𝑀) = (𝐹 ∘ 𝑁))
3	2	fveq1d 6672	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → ((𝐹 ∘ 𝑀)‘𝑅) = ((𝐹 ∘ 𝑁)‘𝑅))
4		cvmliftmolem.4	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝐼 ⊆ (^◡𝑀 “ 𝑊))
5		cvmliftmolem.8	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝑅 ∈ 𝐼)
6	4, 5	sseldd 3968	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 𝑅 ∈ (^◡𝑀 “ 𝑊))
7		cvmliftmoi.m	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ (𝐾 Cn 𝐶))
8		cvmliftmo.y	. . . . . . . . . . . . . . 15 ⊢ 𝑌 = ∪ 𝐾
9		cvmliftmo.b	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = ∪ 𝐶
10	8, 9	cnf 21854	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (𝐾 Cn 𝐶) → 𝑀:𝑌⟶𝐵)
11	7, 10	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀:𝑌⟶𝐵)
12	11	ffnd 6515	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 Fn 𝑌)
13		elpreima 6828	. . . . . . . . . . . 12 ⊢ (𝑀 Fn 𝑌 → (𝑅 ∈ (^◡𝑀 “ 𝑊) ↔ (𝑅 ∈ 𝑌 ∧ (𝑀‘𝑅) ∈ 𝑊)))
14	12, 13	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑅 ∈ (^◡𝑀 “ 𝑊) ↔ (𝑅 ∈ 𝑌 ∧ (𝑀‘𝑅) ∈ 𝑊)))
15	14	simprbda 501	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑅 ∈ (^◡𝑀 “ 𝑊)) → 𝑅 ∈ 𝑌)
16	6, 15	syldan 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑅 ∈ 𝑌)
17		fvco3 6760	. . . . . . . . . 10 ⊢ ((𝑀:𝑌⟶𝐵 ∧ 𝑅 ∈ 𝑌) → ((𝐹 ∘ 𝑀)‘𝑅) = (𝐹‘(𝑀‘𝑅)))
18	11, 17	sylan 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝑌) → ((𝐹 ∘ 𝑀)‘𝑅) = (𝐹‘(𝑀‘𝑅)))
19	16, 18	syldan 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → ((𝐹 ∘ 𝑀)‘𝑅) = (𝐹‘(𝑀‘𝑅)))
20		cvmliftmoi.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (𝐾 Cn 𝐶))
21	8, 9	cnf 21854	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (𝐾 Cn 𝐶) → 𝑁:𝑌⟶𝐵)
22	20, 21	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁:𝑌⟶𝐵)
23		fvco3 6760	. . . . . . . . . 10 ⊢ ((𝑁:𝑌⟶𝐵 ∧ 𝑅 ∈ 𝑌) → ((𝐹 ∘ 𝑁)‘𝑅) = (𝐹‘(𝑁‘𝑅)))
24	22, 23	sylan 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑅 ∈ 𝑌) → ((𝐹 ∘ 𝑁)‘𝑅) = (𝐹‘(𝑁‘𝑅)))
25	16, 24	syldan 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → ((𝐹 ∘ 𝑁)‘𝑅) = (𝐹‘(𝑁‘𝑅)))
26	3, 19, 25	3eqtr3d 2864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝐹‘(𝑀‘𝑅)) = (𝐹‘(𝑁‘𝑅)))
27	26	adantr 483	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝐹‘(𝑀‘𝑅)) = (𝐹‘(𝑁‘𝑅)))
28	14	simplbda 502	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑅 ∈ (^◡𝑀 “ 𝑊)) → (𝑀‘𝑅) ∈ 𝑊)
29	6, 28	syldan 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑀‘𝑅) ∈ 𝑊)
30	29	adantr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝑀‘𝑅) ∈ 𝑊)
31		fvres 6689	. . . . . . 7 ⊢ ((𝑀‘𝑅) ∈ 𝑊 → ((𝐹 ↾ 𝑊)‘(𝑀‘𝑅)) = (𝐹‘(𝑀‘𝑅)))
32	30, 31	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → ((𝐹 ↾ 𝑊)‘(𝑀‘𝑅)) = (𝐹‘(𝑀‘𝑅)))
33	5	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → 𝑅 ∈ 𝐼)
34		fvres 6689	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝐼 → ((𝑁 ↾ 𝐼)‘𝑅) = (𝑁‘𝑅))
35	33, 34	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → ((𝑁 ↾ 𝐼)‘𝑅) = (𝑁‘𝑅))
36		eqid 2821	. . . . . . . . . . 11 ⊢ ∪ (𝐾 ↾_t 𝐼) = ∪ (𝐾 ↾_t 𝐼)
37		cvmliftmolem.5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (𝐾 ↾_t 𝐼) ∈ Conn)
38	37	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝐾 ↾_t 𝐼) ∈ Conn)
39	20	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ (𝐾 Cn 𝐶))
40		cnvimass 5949	. . . . . . . . . . . . . . . . 17 ⊢ (^◡𝑀 “ 𝑊) ⊆ dom 𝑀
41	40, 11	fssdm 6530	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (^◡𝑀 “ 𝑊) ⊆ 𝑌)
42	41	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → (^◡𝑀 “ 𝑊) ⊆ 𝑌)
43	4, 42	sstrd 3977	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝐼 ⊆ 𝑌)
44	8	cnrest 21893	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (𝐾 Cn 𝐶) ∧ 𝐼 ⊆ 𝑌) → (𝑁 ↾ 𝐼) ∈ ((𝐾 ↾_t 𝐼) Cn 𝐶))
45	39, 43, 44	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (𝑁 ↾ 𝐼) ∈ ((𝐾 ↾_t 𝐼) Cn 𝐶))
46		cvmliftmo.f	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
47	46	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → 𝐹 ∈ (𝐶 CovMap 𝐽))
48		cvmtop1 32507	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
49	47, 48	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ Top)
50	9	toptopon 21525	. . . . . . . . . . . . . . 15 ⊢ (𝐶 ∈ Top ↔ 𝐶 ∈ (TopOn‘𝐵))
51	49, 50	sylib 220	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ (TopOn‘𝐵))
52		df-ima 5568	. . . . . . . . . . . . . . 15 ⊢ (𝑁 “ 𝐼) = ran (𝑁 ↾ 𝐼)
53		cvmliftmolem.3	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ∈ 𝑇)
54		elssuni 4868	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑊 ∈ 𝑇 → 𝑊 ⊆ ∪ 𝑇)
55	53, 54	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ ∪ 𝑇)
56		cvmliftmolem.2	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝜓) → 𝑇 ∈ (𝑆‘𝑈))
57		cvmliftmolem.1	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (^◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾_t 𝑢)Homeo(𝐽 ↾_t 𝑘))))})
58	57	cvmsuni 32516	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑇 ∈ (𝑆‘𝑈) → ∪ 𝑇 = (^◡𝐹 “ 𝑈))
59	56, 58	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝜓) → ∪ 𝑇 = (^◡𝐹 “ 𝑈))
60	55, 59	sseqtrd 4007	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ⊆ (^◡𝐹 “ 𝑈))
61		imass2 5965	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 ⊆ (^◡𝐹 “ 𝑈) → (^◡𝑀 “ 𝑊) ⊆ (^◡𝑀 “ (^◡𝐹 “ 𝑈)))
62	60, 61	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → (^◡𝑀 “ 𝑊) ⊆ (^◡𝑀 “ (^◡𝐹 “ 𝑈)))
63	4, 62	sstrd 3977	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → 𝐼 ⊆ (^◡𝑀 “ (^◡𝐹 “ 𝑈)))
64	2	cnveqd 5746	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝜓) → ^◡(𝐹 ∘ 𝑀) = ^◡(𝐹 ∘ 𝑁))
65		cnvco 5756	. . . . . . . . . . . . . . . . . . . 20 ⊢ ^◡(𝐹 ∘ 𝑀) = (^◡𝑀 ∘ ^◡𝐹)
66		cnvco 5756	. . . . . . . . . . . . . . . . . . . 20 ⊢ ^◡(𝐹 ∘ 𝑁) = (^◡𝑁 ∘ ^◡𝐹)
67	64, 65, 66	3eqtr3g 2879	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → (^◡𝑀 ∘ ^◡𝐹) = (^◡𝑁 ∘ ^◡𝐹))
68	67	imaeq1d 5928	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → ((^◡𝑀 ∘ ^◡𝐹) “ 𝑈) = ((^◡𝑁 ∘ ^◡𝐹) “ 𝑈))
69		imaco 6104	. . . . . . . . . . . . . . . . . 18 ⊢ ((^◡𝑀 ∘ ^◡𝐹) “ 𝑈) = (^◡𝑀 “ (^◡𝐹 “ 𝑈))
70		imaco 6104	. . . . . . . . . . . . . . . . . 18 ⊢ ((^◡𝑁 ∘ ^◡𝐹) “ 𝑈) = (^◡𝑁 “ (^◡𝐹 “ 𝑈))
71	68, 69, 70	3eqtr3g 2879	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → (^◡𝑀 “ (^◡𝐹 “ 𝑈)) = (^◡𝑁 “ (^◡𝐹 “ 𝑈)))
72	63, 71	sseqtrd 4007	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → 𝐼 ⊆ (^◡𝑁 “ (^◡𝐹 “ 𝑈)))
73	22	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → 𝑁:𝑌⟶𝐵)
74	73	ffund 6518	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → Fun 𝑁)
75	73	fdmd 6523	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → dom 𝑁 = 𝑌)
76	43, 75	sseqtrrd 4008	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → 𝐼 ⊆ dom 𝑁)
77		funimass3 6824	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝑁 ∧ 𝐼 ⊆ dom 𝑁) → ((𝑁 “ 𝐼) ⊆ (^◡𝐹 “ 𝑈) ↔ 𝐼 ⊆ (^◡𝑁 “ (^◡𝐹 “ 𝑈))))
78	74, 76, 77	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → ((𝑁 “ 𝐼) ⊆ (^◡𝐹 “ 𝑈) ↔ 𝐼 ⊆ (^◡𝑁 “ (^◡𝐹 “ 𝑈))))
79	72, 78	mpbird 259	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → (𝑁 “ 𝐼) ⊆ (^◡𝐹 “ 𝑈))
80	52, 79	eqsstrrid 4016	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → ran (𝑁 ↾ 𝐼) ⊆ (^◡𝐹 “ 𝑈))
81		cnvimass 5949	. . . . . . . . . . . . . . 15 ⊢ (^◡𝐹 “ 𝑈) ⊆ dom 𝐹
82		cvmcn 32509	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐹 ∈ (𝐶 Cn 𝐽))
83	46, 82	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Cn 𝐽))
84		eqid 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ ∪ 𝐽 = ∪ 𝐽
85	9, 84	cnf 21854	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (𝐶 Cn 𝐽) → 𝐹:𝐵⟶∪ 𝐽)
86	83, 85	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝐵⟶∪ 𝐽)
87	86	fdmd 6523	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝐹 = 𝐵)
88	87	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → dom 𝐹 = 𝐵)
89	81, 88	sseqtrid 4019	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (^◡𝐹 “ 𝑈) ⊆ 𝐵)
90		cnrest2 21894	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ (TopOn‘𝐵) ∧ ran (𝑁 ↾ 𝐼) ⊆ (^◡𝐹 “ 𝑈) ∧ (^◡𝐹 “ 𝑈) ⊆ 𝐵) → ((𝑁 ↾ 𝐼) ∈ ((𝐾 ↾_t 𝐼) Cn 𝐶) ↔ (𝑁 ↾ 𝐼) ∈ ((𝐾 ↾_t 𝐼) Cn (𝐶 ↾_t (^◡𝐹 “ 𝑈)))))
91	51, 80, 89, 90	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ((𝑁 ↾ 𝐼) ∈ ((𝐾 ↾_t 𝐼) Cn 𝐶) ↔ (𝑁 ↾ 𝐼) ∈ ((𝐾 ↾_t 𝐼) Cn (𝐶 ↾_t (^◡𝐹 “ 𝑈)))))
92	45, 91	mpbid 234	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (𝑁 ↾ 𝐼) ∈ ((𝐾 ↾_t 𝐼) Cn (𝐶 ↾_t (^◡𝐹 “ 𝑈))))
93	92	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝑁 ↾ 𝐼) ∈ ((𝐾 ↾_t 𝐼) Cn (𝐶 ↾_t (^◡𝐹 “ 𝑈))))
94		df-ss 3952	. . . . . . . . . . . . . 14 ⊢ (𝑊 ⊆ (^◡𝐹 “ 𝑈) ↔ (𝑊 ∩ (^◡𝐹 “ 𝑈)) = 𝑊)
95	60, 94	sylib 220	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (𝑊 ∩ (^◡𝐹 “ 𝑈)) = 𝑊)
96	9	topopn 21514	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ Top → 𝐵 ∈ 𝐶)
97	49, 96	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ 𝐶)
98	97, 89	ssexd 5228	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (^◡𝐹 “ 𝑈) ∈ V)
99	57	cvmsss 32514	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑇 ⊆ 𝐶)
100	56, 99	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝑇 ⊆ 𝐶)
101	100, 53	sseldd 3968	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ∈ 𝐶)
102		elrestr 16702	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ Top ∧ (^◡𝐹 “ 𝑈) ∈ V ∧ 𝑊 ∈ 𝐶) → (𝑊 ∩ (^◡𝐹 “ 𝑈)) ∈ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
103	49, 98, 101, 102	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (𝑊 ∩ (^◡𝐹 “ 𝑈)) ∈ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
104	95, 103	eqeltrrd 2914	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ∈ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
105	104	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → 𝑊 ∈ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
106	57	cvmscld 32520	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝑊 ∈ 𝑇) → 𝑊 ∈ (Clsd‘(𝐶 ↾_t (^◡𝐹 “ 𝑈))))
107	47, 56, 53, 106	syl3anc 1367	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 𝑊 ∈ (Clsd‘(𝐶 ↾_t (^◡𝐹 “ 𝑈))))
108	107	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → 𝑊 ∈ (Clsd‘(𝐶 ↾_t (^◡𝐹 “ 𝑈))))
109		cvmliftmolem.7	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝑄 ∈ 𝐼)
110		cvmliftmo.k	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ Conn)
111		conntop 22025	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ Conn → 𝐾 ∈ Top)
112	110, 111	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ Top)
113	112	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝐾 ∈ Top)
114	8	restuni 21770	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Top ∧ 𝐼 ⊆ 𝑌) → 𝐼 = ∪ (𝐾 ↾_t 𝐼))
115	113, 43, 114	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝐼 = ∪ (𝐾 ↾_t 𝐼))
116	109, 115	eleqtrd 2915	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 𝑄 ∈ ∪ (𝐾 ↾_t 𝐼))
117	116	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → 𝑄 ∈ ∪ (𝐾 ↾_t 𝐼))
118	109	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → 𝑄 ∈ 𝐼)
119		fvres 6689	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ 𝐼 → ((𝑁 ↾ 𝐼)‘𝑄) = (𝑁‘𝑄))
120	118, 119	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → ((𝑁 ↾ 𝐼)‘𝑄) = (𝑁‘𝑄))
121		simpr 487	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝑀‘𝑄) = (𝑁‘𝑄))
122	4, 109	sseldd 3968	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝑄 ∈ (^◡𝑀 “ 𝑊))
123		elpreima 6828	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 Fn 𝑌 → (𝑄 ∈ (^◡𝑀 “ 𝑊) ↔ (𝑄 ∈ 𝑌 ∧ (𝑀‘𝑄) ∈ 𝑊)))
124	12, 123	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑄 ∈ (^◡𝑀 “ 𝑊) ↔ (𝑄 ∈ 𝑌 ∧ (𝑀‘𝑄) ∈ 𝑊)))
125	124	simplbda 502	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑄 ∈ (^◡𝑀 “ 𝑊)) → (𝑀‘𝑄) ∈ 𝑊)
126	122, 125	syldan 593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (𝑀‘𝑄) ∈ 𝑊)
127	126	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝑀‘𝑄) ∈ 𝑊)
128	121, 127	eqeltrrd 2914	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝑁‘𝑄) ∈ 𝑊)
129	120, 128	eqeltrd 2913	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → ((𝑁 ↾ 𝐼)‘𝑄) ∈ 𝑊)
130	36, 38, 93, 105, 108, 117, 129	conncn 22034	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝑁 ↾ 𝐼):∪ (𝐾 ↾_t 𝐼)⟶𝑊)
131	115	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → 𝐼 = ∪ (𝐾 ↾_t 𝐼))
132	131	feq2d 6500	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → ((𝑁 ↾ 𝐼):𝐼⟶𝑊 ↔ (𝑁 ↾ 𝐼):∪ (𝐾 ↾_t 𝐼)⟶𝑊))
133	130, 132	mpbird 259	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝑁 ↾ 𝐼):𝐼⟶𝑊)
134	133, 33	ffvelrnd 6852	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → ((𝑁 ↾ 𝐼)‘𝑅) ∈ 𝑊)
135	35, 134	eqeltrrd 2914	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝑁‘𝑅) ∈ 𝑊)
136		fvres 6689	. . . . . . 7 ⊢ ((𝑁‘𝑅) ∈ 𝑊 → ((𝐹 ↾ 𝑊)‘(𝑁‘𝑅)) = (𝐹‘(𝑁‘𝑅)))
137	135, 136	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → ((𝐹 ↾ 𝑊)‘(𝑁‘𝑅)) = (𝐹‘(𝑁‘𝑅)))
138	27, 32, 137	3eqtr4d 2866	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → ((𝐹 ↾ 𝑊)‘(𝑀‘𝑅)) = ((𝐹 ↾ 𝑊)‘(𝑁‘𝑅)))
139	57	cvmsf1o 32519	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝑊 ∈ 𝑇) → (𝐹 ↾ 𝑊):𝑊–1-1-onto→𝑈)
140	47, 56, 53, 139	syl3anc 1367	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝐹 ↾ 𝑊):𝑊–1-1-onto→𝑈)
141		f1of1 6614	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝑊):𝑊–1-1-onto→𝑈 → (𝐹 ↾ 𝑊):𝑊–1-1→𝑈)
142	140, 141	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝐹 ↾ 𝑊):𝑊–1-1→𝑈)
143	142	adantr 483	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝐹 ↾ 𝑊):𝑊–1-1→𝑈)
144		f1fveq 7020	. . . . . 6 ⊢ (((𝐹 ↾ 𝑊):𝑊–1-1→𝑈 ∧ ((𝑀‘𝑅) ∈ 𝑊 ∧ (𝑁‘𝑅) ∈ 𝑊)) → (((𝐹 ↾ 𝑊)‘(𝑀‘𝑅)) = ((𝐹 ↾ 𝑊)‘(𝑁‘𝑅)) ↔ (𝑀‘𝑅) = (𝑁‘𝑅)))
145	143, 30, 135, 144	syl12anc 834	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (((𝐹 ↾ 𝑊)‘(𝑀‘𝑅)) = ((𝐹 ↾ 𝑊)‘(𝑁‘𝑅)) ↔ (𝑀‘𝑅) = (𝑁‘𝑅)))
146	138, 145	mpbid 234	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑀‘𝑄) = (𝑁‘𝑄)) → (𝑀‘𝑅) = (𝑁‘𝑅))
147	146	ex 415	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀‘𝑄) = (𝑁‘𝑄) → (𝑀‘𝑅) = (𝑁‘𝑅)))
148	124	simprbda 501	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 ∈ (^◡𝑀 “ 𝑊)) → 𝑄 ∈ 𝑌)
149	122, 148	syldan 593	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑄 ∈ 𝑌)
150		fveq2 6670	. . . . . 6 ⊢ (𝑥 = 𝑄 → (𝑀‘𝑥) = (𝑀‘𝑄))
151		fveq2 6670	. . . . . 6 ⊢ (𝑥 = 𝑄 → (𝑁‘𝑥) = (𝑁‘𝑄))
152	150, 151	eqeq12d 2837	. . . . 5 ⊢ (𝑥 = 𝑄 → ((𝑀‘𝑥) = (𝑁‘𝑥) ↔ (𝑀‘𝑄) = (𝑁‘𝑄)))
153	152	elrab3 3681	. . . 4 ⊢ (𝑄 ∈ 𝑌 → (𝑄 ∈ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)} ↔ (𝑀‘𝑄) = (𝑁‘𝑄)))
154	149, 153	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑄 ∈ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)} ↔ (𝑀‘𝑄) = (𝑁‘𝑄)))
155		fveq2 6670	. . . . . 6 ⊢ (𝑥 = 𝑅 → (𝑀‘𝑥) = (𝑀‘𝑅))
156		fveq2 6670	. . . . . 6 ⊢ (𝑥 = 𝑅 → (𝑁‘𝑥) = (𝑁‘𝑅))
157	155, 156	eqeq12d 2837	. . . . 5 ⊢ (𝑥 = 𝑅 → ((𝑀‘𝑥) = (𝑁‘𝑥) ↔ (𝑀‘𝑅) = (𝑁‘𝑅)))
158	157	elrab3 3681	. . . 4 ⊢ (𝑅 ∈ 𝑌 → (𝑅 ∈ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)} ↔ (𝑀‘𝑅) = (𝑁‘𝑅)))
159	16, 158	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑅 ∈ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)} ↔ (𝑀‘𝑅) = (𝑁‘𝑅)))
160	147, 154, 159	3imtr4d 296	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑄 ∈ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)} → 𝑅 ∈ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)}))
161	22	ffnd 6515	. . . . 5 ⊢ (𝜑 → 𝑁 Fn 𝑌)
162		fndmin 6815	. . . . 5 ⊢ ((𝑀 Fn 𝑌 ∧ 𝑁 Fn 𝑌) → dom (𝑀 ∩ 𝑁) = {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)})
163	12, 161, 162	syl2anc 586	. . . 4 ⊢ (𝜑 → dom (𝑀 ∩ 𝑁) = {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)})
164	163	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝜓) → dom (𝑀 ∩ 𝑁) = {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)})
165	164	eleq2d 2898	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑄 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑄 ∈ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)}))
166	164	eleq2d 2898	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑅 ∈ dom (𝑀 ∩ 𝑁) ↔ 𝑅 ∈ {𝑥 ∈ 𝑌 ∣ (𝑀‘𝑥) = (𝑁‘𝑥)}))
167	160, 165, 166	3imtr4d 296	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑄 ∈ dom (𝑀 ∩ 𝑁) → 𝑅 ∈ dom (𝑀 ∩ 𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 {crab 3142 Vcvv 3494 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 {csn 4567 ∪ cuni 4838 ↦ cmpt 5146 ^◡ccnv 5554 dom cdm 5555 ran crn 5556 ↾ cres 5557 “ cima 5558 ∘ ccom 5559 Fun wfun 6349 Fn wfn 6350 ⟶wf 6351 –1-1→wf1 6352 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 ↾_t crest 16694 Topctop 21501 TopOnctopon 21518 Clsdccld 21624 Cn ccn 21832 Conncconn 22019 𝑛-Locally cnlly 22073 Homeochmeo 22361 CovMap ccvm 32502

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-fin 8513 df-fi 8875 df-rest 16696 df-topgen 16717 df-top 21502 df-topon 21519 df-bases 21554 df-cld 21627 df-cn 21835 df-conn 22020 df-hmeo 22363 df-cvm 32503

This theorem is referenced by: cvmliftmolem2 32529

W3C validator