Theorem cvmscld 35765

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 35752	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
2	1	3ad2ant1 1151	. . . . 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 35761	. . . . . . 7 ⊢ (𝑇 ∈ (𝑆‘𝑈) → ∪ 𝑇 = (◡𝐹 “ 𝑈))
5	4	3ad2ant2 1152	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑇 = (◡𝐹 “ 𝑈))
6	3	cvmsss 35759	. . . . . . . 8 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑇 ⊆ 𝐶)
7	6	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝑇 ⊆ 𝐶)
8	7	unissd 4882	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑇 ⊆ ∪ 𝐶)
9	5, 8	eqsstrrd 3972	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (◡𝐹 “ 𝑈) ⊆ ∪ 𝐶)
10		eqid 2763	. . . . . 6 ⊢ ∪ 𝐶 = ∪ 𝐶
11	10	restuni 23319	. . . . 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 4080	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ((◡𝐹 “ 𝑈) ∖ ∪ (𝑇 ∖ {𝐴})) = (∪ (𝐶 ↾t (◡𝐹 “ 𝑈)) ∖ ∪ (𝑇 ∖ {𝐴})))
14		unisng 4890	. . . . . . 7 ⊢ (𝐴 ∈ 𝑇 → ∪ {𝐴} = 𝐴)
15	14	3ad2ant3 1153	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ {𝐴} = 𝐴)
16	15	uneq2d 4122	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∪ ∪ {𝐴}) = (∪ (𝑇 ∖ {𝐴}) ∪ 𝐴))
17		uniun 4895	. . . . . 6 ⊢ ∪ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (∪ (𝑇 ∖ {𝐴}) ∪ ∪ {𝐴})
18		undif1 4437	. . . . . . . . 9 ⊢ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (𝑇 ∪ {𝐴})
19		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝑇)
20	19	snssd 4752	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → {𝐴} ⊆ 𝑇)
21		ssequn2 4142	. . . . . . . . . 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 2810	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = 𝑇)
24	23	unieqd 4885	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = ∪ 𝑇)
25	24, 5	eqtrd 2798	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (◡𝐹 “ 𝑈))
26	17, 25	eqtr3id 2812	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∪ ∪ {𝐴}) = (◡𝐹 “ 𝑈))
27	16, 26	eqtr3d 2800	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∪ 𝐴) = (◡𝐹 “ 𝑈))
28		difss 4090	. . . . . . 7 ⊢ (𝑇 ∖ {𝐴}) ⊆ 𝑇
29	28	unissi 4881	. . . . . 6 ⊢ ∪ (𝑇 ∖ {𝐴}) ⊆ ∪ 𝑇
30	29, 5	sseqtrid 3979	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ (𝑇 ∖ {𝐴}) ⊆ (◡𝐹 “ 𝑈))
31		uniiun 5023	. . . . . . . 8 ⊢ ∪ (𝑇 ∖ {𝐴}) = ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥
32	31	ineq2i 4170	. . . . . . 7 ⊢ (𝐴 ∩ ∪ (𝑇 ∖ {𝐴})) = (𝐴 ∩ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥)
33		incom 4162	. . . . . . 7 ⊢ (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = (𝐴 ∩ ∪ (𝑇 ∖ {𝐴}))
34		iunin2 5035	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥)
35	32, 33, 34	3eqtr4i 2796	. . . . . 6 ⊢ (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥)
36		eldifsn 4753	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑇 ∖ {𝐴}) ↔ (𝑥 ∈ 𝑇 ∧ 𝑥 ≠ 𝐴))
37		nesym 3014	. . . . . . . . . . . 12 ⊢ (𝑥 ≠ 𝐴 ↔ ¬ 𝐴 = 𝑥)
38	3	cvmsdisj 35762	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇 ∧ 𝑥 ∈ 𝑇) → (𝐴 = 𝑥 ∨ (𝐴 ∩ 𝑥) = ∅))
39	38	3expa 1136	. . . . . . . . . . . . 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 4981	. . . . . . . 8 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥) = ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})∅)
45	44	3adant1 1148	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥) = ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})∅)
46		iun0 5026	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})∅ = ∅
47	45, 46	eqtrdi 2814	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥) = ∅)
48	35, 47	eqtrid 2810	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∅)
49		uneqdifeq 4453	. . . . 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 2800	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝐶 ↾t (◡𝐹 “ 𝑈)) ∖ ∪ (𝑇 ∖ {𝐴})) = 𝐴)
53		uniexg 7738	. . . . . 6 ⊢ (𝑇 ∈ (𝑆‘𝑈) → ∪ 𝑇 ∈ V)
54	53	3ad2ant2 1152	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑇 ∈ V)
55	5, 54	eqeltrrd 2864	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (◡𝐹 “ 𝑈) ∈ V)
56		resttop 23317	. . . 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 4904	. . . . . . . . . . 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 3973	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝑥 ⊆ (◡𝐹 “ 𝑈))
62		dfss2 3923	. . . . . . . . 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 3937	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝐶)
67		elrestr 17476	. . . . . . . . 9 ⊢ ((𝐶 ∈ Top ∧ (◡𝐹 “ 𝑈) ∈ V ∧ 𝑥 ∈ 𝐶) → (𝑥 ∩ (◡𝐹 “ 𝑈)) ∈ (𝐶 ↾t (◡𝐹 “ 𝑈)))
68	64, 65, 66, 67	syl3anc 1398	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → (𝑥 ∩ (◡𝐹 “ 𝑈)) ∈ (𝐶 ↾t (◡𝐹 “ 𝑈)))
69	63, 68	eqeltrrd 2864	. . . . . . 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 3943	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝑇 ⊆ (𝐶 ↾t (◡𝐹 “ 𝑈)))
72	71	ssdifssd 4101	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝑇 ∖ {𝐴}) ⊆ (𝐶 ↾t (◡𝐹 “ 𝑈)))
73		uniopn 23054	. . . 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 2763	. . . 4 ⊢ ∪ (𝐶 ↾t (◡𝐹 “ 𝑈)) = ∪ (𝐶 ↾t (◡𝐹 “ 𝑈))
76	75	opncld 23190	. . 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 2864	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 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  {crab 3416  Vcvv 3455   ∖ cdif 3902   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4562  {csn 4589  ∪ cuni 4872  ∪ ciun 4956   ↦ cmpt 5192  ^◡ccnv 5660   ↾ cres 5663   “ cima 5664  ‘cfv 6536  (class class class)co 7410   ↾_t crest 17468  Topctop 23050  Clsdccld 23173  Homeochmeo 23910   CovMap ccvm 35747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-en 8940  df-fin 8943  df-fi 9367  df-rest 17470  df-topgen 17491  df-top 23051  df-topon 23068  df-bases 23103  df-cld 23176  df-cvm 35748

This theorem is referenced by:  cvmliftmolem1  35773  cvmlift2lem9  35803  cvmlift3lem6  35816