Theorem cvmscld 35664

Description: The sets of an even covering are clopen in the subspace topology on 𝑇. (Contributed by Mario Carneiro, 14-Feb-2015.)

Hypothesis

Ref	Expression
cvmcov.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))))})

Assertion

Ref	Expression
cvmscld	⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ (Clsd‘(𝐶 ↾t (◡𝐹 “ 𝑈))))

Distinct variable groups:   𝑘,𝑠,𝑢,𝑣,𝐶   𝑘,𝐹,𝑠,𝑢,𝑣   𝑘,𝐽,𝑠,𝑢,𝑣   𝑈,𝑘,𝑠,𝑢,𝑣   𝑇,𝑠,𝑢,𝑣   𝑢,𝐴,𝑣

Allowed substitution hints:   𝐴(𝑘,𝑠)   𝑆(𝑣,𝑢,𝑘,𝑠)   𝑇(𝑘)

Proof of Theorem cvmscld

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmtop1 35651	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
2	1	3ad2ant1 1149	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐶 ∈ Top)
3		cvmcov.1	. . . . . . . 8 ⊢ 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))))})
4	3	cvmsuni 35660	. . . . . . 7 ⊢ (𝑇 ∈ (𝑆‘𝑈) → ∪ 𝑇 = (◡𝐹 “ 𝑈))
5	4	3ad2ant2 1150	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑇 = (◡𝐹 “ 𝑈))
6	3	cvmsss 35658	. . . . . . . 8 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑇 ⊆ 𝐶)
7	6	3ad2ant2 1150	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝑇 ⊆ 𝐶)
8	7	unissd 4886	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑇 ⊆ ∪ 𝐶)
9	5, 8	eqsstrrd 3980	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (◡𝐹 “ 𝑈) ⊆ ∪ 𝐶)
10		eqid 2769	. . . . . 6 ⊢ ∪ 𝐶 = ∪ 𝐶
11	10	restuni 23288	. . . . 5 ⊢ ((𝐶 ∈ Top ∧ (◡𝐹 “ 𝑈) ⊆ ∪ 𝐶) → (◡𝐹 “ 𝑈) = ∪ (𝐶 ↾t (◡𝐹 “ 𝑈)))
12	2, 9, 11	syl2anc 595	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (◡𝐹 “ 𝑈) = ∪ (𝐶 ↾t (◡𝐹 “ 𝑈)))
13	12	difeq1d 4088	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ((◡𝐹 “ 𝑈) ∖ ∪ (𝑇 ∖ {𝐴})) = (∪ (𝐶 ↾t (◡𝐹 “ 𝑈)) ∖ ∪ (𝑇 ∖ {𝐴})))
14		unisng 4894	. . . . . . 7 ⊢ (𝐴 ∈ 𝑇 → ∪ {𝐴} = 𝐴)
15	14	3ad2ant3 1151	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ {𝐴} = 𝐴)
16	15	uneq2d 4130	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∪ ∪ {𝐴}) = (∪ (𝑇 ∖ {𝐴}) ∪ 𝐴))
17		uniun 4899	. . . . . 6 ⊢ ∪ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (∪ (𝑇 ∖ {𝐴}) ∪ ∪ {𝐴})
18		undif1 4442	. . . . . . . . 9 ⊢ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (𝑇 ∪ {𝐴})
19		simp3 1154	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝑇)
20	19	snssd 4757	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → {𝐴} ⊆ 𝑇)
21		ssequn2 4150	. . . . . . . . . 10 ⊢ ({𝐴} ⊆ 𝑇 ↔ (𝑇 ∪ {𝐴}) = 𝑇)
22	20, 21	sylib 221	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝑇 ∪ {𝐴}) = 𝑇)
23	18, 22	eqtrid 2816	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = 𝑇)
24	23	unieqd 4889	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = ∪ 𝑇)
25	24, 5	eqtrd 2804	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (◡𝐹 “ 𝑈))
26	17, 25	eqtr3id 2818	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∪ ∪ {𝐴}) = (◡𝐹 “ 𝑈))
27	16, 26	eqtr3d 2806	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∪ 𝐴) = (◡𝐹 “ 𝑈))
28		difss 4098	. . . . . . 7 ⊢ (𝑇 ∖ {𝐴}) ⊆ 𝑇
29	28	unissi 4885	. . . . . 6 ⊢ ∪ (𝑇 ∖ {𝐴}) ⊆ ∪ 𝑇
30	29, 5	sseqtrid 3987	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ (𝑇 ∖ {𝐴}) ⊆ (◡𝐹 “ 𝑈))
31		uniiun 5027	. . . . . . . 8 ⊢ ∪ (𝑇 ∖ {𝐴}) = ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥
32	31	ineq2i 4178	. . . . . . 7 ⊢ (𝐴 ∩ ∪ (𝑇 ∖ {𝐴})) = (𝐴 ∩ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥)
33		incom 4170	. . . . . . 7 ⊢ (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = (𝐴 ∩ ∪ (𝑇 ∖ {𝐴}))
34		iunin2 5039	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥)
35	32, 33, 34	3eqtr4i 2802	. . . . . 6 ⊢ (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥)
36		eldifsn 4758	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑇 ∖ {𝐴}) ↔ (𝑥 ∈ 𝑇 ∧ 𝑥 ≠ 𝐴))
37		nesym 3020	. . . . . . . . . . . 12 ⊢ (𝑥 ≠ 𝐴 ↔ ¬ 𝐴 = 𝑥)
38	3	cvmsdisj 35661	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇 ∧ 𝑥 ∈ 𝑇) → (𝐴 = 𝑥 ∨ (𝐴 ∩ 𝑥) = ∅))
39	38	3expa 1134	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → (𝐴 = 𝑥 ∨ (𝐴 ∩ 𝑥) = ∅))
40	39	ord 877	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → (¬ 𝐴 = 𝑥 → (𝐴 ∩ 𝑥) = ∅))
41	37, 40	biimtrid 245	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → (𝑥 ≠ 𝐴 → (𝐴 ∩ 𝑥) = ∅))
42	41	impr 459	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ (𝑥 ∈ 𝑇 ∧ 𝑥 ≠ 𝐴)) → (𝐴 ∩ 𝑥) = ∅)
43	36, 42	sylan2b 605	. . . . . . . . 9 ⊢ (((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ (𝑇 ∖ {𝐴})) → (𝐴 ∩ 𝑥) = ∅)
44	43	iuneq2dv 4985	. . . . . . . 8 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥) = ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})∅)
45	44	3adant1 1146	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥) = ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})∅)
46		iun0 5030	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})∅ = ∅
47	45, 46	eqtrdi 2820	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥) = ∅)
48	35, 47	eqtrid 2816	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∅)
49		uneqdifeq 4458	. . . . 5 ⊢ ((∪ (𝑇 ∖ {𝐴}) ⊆ (◡𝐹 “ 𝑈) ∧ (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∅) → ((∪ (𝑇 ∖ {𝐴}) ∪ 𝐴) = (◡𝐹 “ 𝑈) ↔ ((◡𝐹 “ 𝑈) ∖ ∪ (𝑇 ∖ {𝐴})) = 𝐴))
50	30, 48, 49	syl2anc 595	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ((∪ (𝑇 ∖ {𝐴}) ∪ 𝐴) = (◡𝐹 “ 𝑈) ↔ ((◡𝐹 “ 𝑈) ∖ ∪ (𝑇 ∖ {𝐴})) = 𝐴))
51	27, 50	mpbid 235	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ((◡𝐹 “ 𝑈) ∖ ∪ (𝑇 ∖ {𝐴})) = 𝐴)
52	13, 51	eqtr3d 2806	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝐶 ↾t (◡𝐹 “ 𝑈)) ∖ ∪ (𝑇 ∖ {𝐴})) = 𝐴)
53		uniexg 7739	. . . . . 6 ⊢ (𝑇 ∈ (𝑆‘𝑈) → ∪ 𝑇 ∈ V)
54	53	3ad2ant2 1150	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑇 ∈ V)
55	5, 54	eqeltrrd 2870	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (◡𝐹 “ 𝑈) ∈ V)
56		resttop 23286	. . . 4 ⊢ ((𝐶 ∈ Top ∧ (◡𝐹 “ 𝑈) ∈ V) → (𝐶 ↾t (◡𝐹 “ 𝑈)) ∈ Top)
57	2, 55, 56	syl2anc 595	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐶 ↾t (◡𝐹 “ 𝑈)) ∈ Top)
58		elssuni 4908	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑇 → 𝑥 ⊆ ∪ 𝑇)
59	58	adantl 486	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝑥 ⊆ ∪ 𝑇)
60	5	adantr 485	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → ∪ 𝑇 = (◡𝐹 “ 𝑈))
61	59, 60	sseqtrd 3981	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝑥 ⊆ (◡𝐹 “ 𝑈))
62		dfss2 3931	. . . . . . . . 9 ⊢ (𝑥 ⊆ (◡𝐹 “ 𝑈) ↔ (𝑥 ∩ (◡𝐹 “ 𝑈)) = 𝑥)
63	61, 62	sylib 221	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → (𝑥 ∩ (◡𝐹 “ 𝑈)) = 𝑥)
64	2	adantr 485	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝐶 ∈ Top)
65	55	adantr 485	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → (◡𝐹 “ 𝑈) ∈ V)
66	7	sselda 3945	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝐶)
67		elrestr 17481	. . . . . . . . 9 ⊢ ((𝐶 ∈ Top ∧ (◡𝐹 “ 𝑈) ∈ V ∧ 𝑥 ∈ 𝐶) → (𝑥 ∩ (◡𝐹 “ 𝑈)) ∈ (𝐶 ↾t (◡𝐹 “ 𝑈)))
68	64, 65, 66, 67	syl3anc 1396	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → (𝑥 ∩ (◡𝐹 “ 𝑈)) ∈ (𝐶 ↾t (◡𝐹 “ 𝑈)))
69	63, 68	eqeltrrd 2870	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ (𝐶 ↾t (◡𝐹 “ 𝑈)))
70	69	ex 417	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝑥 ∈ 𝑇 → 𝑥 ∈ (𝐶 ↾t (◡𝐹 “ 𝑈))))
71	70	ssrdv 3951	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝑇 ⊆ (𝐶 ↾t (◡𝐹 “ 𝑈)))
72	71	ssdifssd 4109	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝑇 ∖ {𝐴}) ⊆ (𝐶 ↾t (◡𝐹 “ 𝑈)))
73		uniopn 23023	. . . 4 ⊢ (((𝐶 ↾t (◡𝐹 “ 𝑈)) ∈ Top ∧ (𝑇 ∖ {𝐴}) ⊆ (𝐶 ↾t (◡𝐹 “ 𝑈))) → ∪ (𝑇 ∖ {𝐴}) ∈ (𝐶 ↾t (◡𝐹 “ 𝑈)))
74	57, 72, 73	syl2anc 595	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ (𝑇 ∖ {𝐴}) ∈ (𝐶 ↾t (◡𝐹 “ 𝑈)))
75		eqid 2769	. . . 4 ⊢ ∪ (𝐶 ↾t (◡𝐹 “ 𝑈)) = ∪ (𝐶 ↾t (◡𝐹 “ 𝑈))
76	75	opncld 23159	. . 3 ⊢ (((𝐶 ↾t (◡𝐹 “ 𝑈)) ∈ Top ∧ ∪ (𝑇 ∖ {𝐴}) ∈ (𝐶 ↾t (◡𝐹 “ 𝑈))) → (∪ (𝐶 ↾t (◡𝐹 “ 𝑈)) ∖ ∪ (𝑇 ∖ {𝐴})) ∈ (Clsd‘(𝐶 ↾t (◡𝐹 “ 𝑈))))
77	57, 74, 76	syl2anc 595	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝐶 ↾t (◡𝐹 “ 𝑈)) ∖ ∪ (𝑇 ∖ {𝐴})) ∈ (Clsd‘(𝐶 ↾t (◡𝐹 “ 𝑈))))
78	52, 77	eqeltrrd 2870	1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ (Clsd‘(𝐶 ↾t (◡𝐹 “ 𝑈))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  {crab 3423  Vcvv 3463   ∖ cdif 3910   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  {csn 4594  ∪ cuni 4876  ∪ ciun 4960   ↦ cmpt 5196  ^◡ccnv 5661   ↾ cres 5664   “ cima 5665  ‘cfv 6537  (class class class)co 7411   ↾_t crest 17473  Topctop 23019  Clsdccld 23142  Homeochmeo 23879   CovMap ccvm 35646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-en 8944  df-fin 8947  df-fi 9371  df-rest 17475  df-topgen 17496  df-top 23020  df-topon 23037  df-bases 23072  df-cld 23145  df-cvm 35647

This theorem is referenced by:  cvmliftmolem1  35672  cvmlift2lem9  35702  cvmlift3lem6  35715