Theorem cvmscld 36038

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 36025	. . . . . 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 36034	. . . . . . 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 36032	. . . . . . . 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 4877	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑇 ⊆ ∪ 𝐶)
9	5, 8	eqsstrrd 3966	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (◡𝐹 “ 𝑈) ⊆ ∪ 𝐶)
10		eqid 2761	. . . . . 6 ⊢ ∪ 𝐶 = ∪ 𝐶
11	10	restuni 23480	. . . . 5 ⊢ ((𝐶 ∈ Top ∧ (◡𝐹 “ 𝑈) ⊆ ∪ 𝐶) → (◡𝐹 “ 𝑈) = ∪ (𝐶 ↾t (◡𝐹 “ 𝑈)))
12	2, 9, 11	syl2anc 596	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (◡𝐹 “ 𝑈) = ∪ (𝐶 ↾t (◡𝐹 “ 𝑈)))
13	12	difeq1d 4073	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ((◡𝐹 “ 𝑈) ∖ ∪ (𝑇 ∖ {𝐴})) = (∪ (𝐶 ↾t (◡𝐹 “ 𝑈)) ∖ ∪ (𝑇 ∖ {𝐴})))
14		unisng 4885	. . . . . . 7 ⊢ (𝐴 ∈ 𝑇 → ∪ {𝐴} = 𝐴)
15	14	3ad2ant3 1153	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ {𝐴} = 𝐴)
16	15	uneq2d 4115	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∪ ∪ {𝐴}) = (∪ (𝑇 ∖ {𝐴}) ∪ 𝐴))
17		uniun 4890	. . . . . 6 ⊢ ∪ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (∪ (𝑇 ∖ {𝐴}) ∪ ∪ {𝐴})
18		undif1 4430	. . . . . . . . 9 ⊢ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (𝑇 ∪ {𝐴})
19		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝑇)
20	19	snssd 4747	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → {𝐴} ⊆ 𝑇)
21		ssequn2 4135	. . . . . . . . . 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 2808	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = 𝑇)
24	23	unieqd 4880	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = ∪ 𝑇)
25	24, 5	eqtrd 2796	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (◡𝐹 “ 𝑈))
26	17, 25	eqtr3id 2810	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∪ ∪ {𝐴}) = (◡𝐹 “ 𝑈))
27	16, 26	eqtr3d 2798	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∪ 𝐴) = (◡𝐹 “ 𝑈))
28		difss 4083	. . . . . . 7 ⊢ (𝑇 ∖ {𝐴}) ⊆ 𝑇
29	28	unissi 4876	. . . . . 6 ⊢ ∪ (𝑇 ∖ {𝐴}) ⊆ ∪ 𝑇
30	29, 5	sseqtrid 3973	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ (𝑇 ∖ {𝐴}) ⊆ (◡𝐹 “ 𝑈))
31		uniiun 5017	. . . . . . . 8 ⊢ ∪ (𝑇 ∖ {𝐴}) = ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥
32	31	ineq2i 4163	. . . . . . 7 ⊢ (𝐴 ∩ ∪ (𝑇 ∖ {𝐴})) = (𝐴 ∩ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥)
33		incom 4155	. . . . . . 7 ⊢ (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = (𝐴 ∩ ∪ (𝑇 ∖ {𝐴}))
34		iunin2 5029	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥)
35	32, 33, 34	3eqtr4i 2794	. . . . . 6 ⊢ (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥)
36		eldifsn 4748	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑇 ∖ {𝐴}) ↔ (𝑥 ∈ 𝑇 ∧ 𝑥 ≠ 𝐴))
37		nesym 3012	. . . . . . . . . . . 12 ⊢ (𝑥 ≠ 𝐴 ↔ ¬ 𝐴 = 𝑥)
38	3	cvmsdisj 36035	. . . . . . . . . . . . . 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 878	. . . . . . . . . . . 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 460	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ (𝑥 ∈ 𝑇 ∧ 𝑥 ≠ 𝐴)) → (𝐴 ∩ 𝑥) = ∅)
43	36, 42	sylan2b 606	. . . . . . . . 9 ⊢ (((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ (𝑇 ∖ {𝐴})) → (𝐴 ∩ 𝑥) = ∅)
44	43	iuneq2dv 4976	. . . . . . . 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 5020	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})∅ = ∅
47	45, 46	eqtrdi 2812	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥) = ∅)
48	35, 47	eqtrid 2808	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∅)
49		uneqdifeq 4448	. . . . 5 ⊢ ((∪ (𝑇 ∖ {𝐴}) ⊆ (◡𝐹 “ 𝑈) ∧ (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∅) → ((∪ (𝑇 ∖ {𝐴}) ∪ 𝐴) = (◡𝐹 “ 𝑈) ↔ ((◡𝐹 “ 𝑈) ∖ ∪ (𝑇 ∖ {𝐴})) = 𝐴))
50	30, 48, 49	syl2anc 596	. . . 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 2798	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝐶 ↾t (◡𝐹 “ 𝑈)) ∖ ∪ (𝑇 ∖ {𝐴})) = 𝐴)
53		uniexg 7757	. . . . . 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 2862	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (◡𝐹 “ 𝑈) ∈ V)
56		resttop 23478	. . . 4 ⊢ ((𝐶 ∈ Top ∧ (◡𝐹 “ 𝑈) ∈ V) → (𝐶 ↾t (◡𝐹 “ 𝑈)) ∈ Top)
57	2, 55, 56	syl2anc 596	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐶 ↾t (◡𝐹 “ 𝑈)) ∈ Top)
58		elssuni 4899	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑇 → 𝑥 ⊆ ∪ 𝑇)
59	58	adantl 487	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝑥 ⊆ ∪ 𝑇)
60	5	adantr 486	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → ∪ 𝑇 = (◡𝐹 “ 𝑈))
61	59, 60	sseqtrd 3967	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝑥 ⊆ (◡𝐹 “ 𝑈))
62		dfss2 3917	. . . . . . . . 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 486	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝐶 ∈ Top)
65	55	adantr 486	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → (◡𝐹 “ 𝑈) ∈ V)
66	7	sselda 3931	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝐶)
67		elrestr 17599	. . . . . . . . 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 2862	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ (𝐶 ↾t (◡𝐹 “ 𝑈)))
70	69	ex 418	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝑥 ∈ 𝑇 → 𝑥 ∈ (𝐶 ↾t (◡𝐹 “ 𝑈))))
71	70	ssrdv 3937	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝑇 ⊆ (𝐶 ↾t (◡𝐹 “ 𝑈)))
72	71	ssdifssd 4094	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝑇 ∖ {𝐴}) ⊆ (𝐶 ↾t (◡𝐹 “ 𝑈)))
73		uniopn 23215	. . . 4 ⊢ (((𝐶 ↾t (◡𝐹 “ 𝑈)) ∈ Top ∧ (𝑇 ∖ {𝐴}) ⊆ (𝐶 ↾t (◡𝐹 “ 𝑈))) → ∪ (𝑇 ∖ {𝐴}) ∈ (𝐶 ↾t (◡𝐹 “ 𝑈)))
74	57, 72, 73	syl2anc 596	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ (𝑇 ∖ {𝐴}) ∈ (𝐶 ↾t (◡𝐹 “ 𝑈)))
75		eqid 2761	. . . 4 ⊢ ∪ (𝐶 ↾t (◡𝐹 “ 𝑈)) = ∪ (𝐶 ↾t (◡𝐹 “ 𝑈))
76	75	opncld 23351	. . 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 596	. 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 2862	1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ (Clsd‘(𝐶 ↾t (◡𝐹 “ 𝑈))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  {crab 3413  Vcvv 3451   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584  ∪ cuni 4867  ∪ ciun 4951   ↦ cmpt 5186  ^◡ccnv 5650   ↾ cres 5653   “ cima 5654  ‘cfv 6538  (class class class)co 7420   ↾_t crest 17591  Topctop 23211  Clsdccld 23334  Homeochmeo 24072   CovMap ccvm 36020

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-en 8974  df-fin 8977  df-fi 9403  df-rest 17593  df-topgen 17614  df-top 23212  df-topon 23229  df-bases 23264  df-cld 23337  df-cvm 36021

This theorem is used by:  cvmliftmolem1  36046  cvmlift2lem9  36076  cvmlift3lem6  36089