cvmscld - Mathbox for Mario Carneiro

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Theorem cvmscld 35623

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 35610	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
2	1	3ad2ant1 1146	. . . . 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 35619	. . . . . . 7 ⊢ (𝑇 ∈ (𝑆‘𝑈) → ∪ 𝑇 = (^◡𝐹 “ 𝑈))
5	4	3ad2ant2 1147	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑇 = (^◡𝐹 “ 𝑈))
6	3	cvmsss 35617	. . . . . . . 8 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑇 ⊆ 𝐶)
7	6	3ad2ant2 1147	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝑇 ⊆ 𝐶)
8	7	unissd 4875	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑇 ⊆ ∪ 𝐶)
9	5, 8	eqsstrrd 3971	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (^◡𝐹 “ 𝑈) ⊆ ∪ 𝐶)
10		eqid 2762	. . . . . 6 ⊢ ∪ 𝐶 = ∪ 𝐶
11	10	restuni 23222	. . . . 5 ⊢ ((𝐶 ∈ Top ∧ (^◡𝐹 “ 𝑈) ⊆ ∪ 𝐶) → (^◡𝐹 “ 𝑈) = ∪ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
12	2, 9, 11	syl2anc 593	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (^◡𝐹 “ 𝑈) = ∪ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
13	12	difeq1d 4079	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ((^◡𝐹 “ 𝑈) ∖ ∪ (𝑇 ∖ {𝐴})) = (∪ (𝐶 ↾_t (^◡𝐹 “ 𝑈)) ∖ ∪ (𝑇 ∖ {𝐴})))
14		unisng 4883	. . . . . . 7 ⊢ (𝐴 ∈ 𝑇 → ∪ {𝐴} = 𝐴)
15	14	3ad2ant3 1148	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ {𝐴} = 𝐴)
16	15	uneq2d 4121	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∪ ∪ {𝐴}) = (∪ (𝑇 ∖ {𝐴}) ∪ 𝐴))
17		uniun 4888	. . . . . 6 ⊢ ∪ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (∪ (𝑇 ∖ {𝐴}) ∪ ∪ {𝐴})
18		undif1 4430	. . . . . . . . 9 ⊢ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (𝑇 ∪ {𝐴})
19		simp3 1151	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝑇)
20	19	snssd 4745	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → {𝐴} ⊆ 𝑇)
21		ssequn2 4141	. . . . . . . . . 10 ⊢ ({𝐴} ⊆ 𝑇 ↔ (𝑇 ∪ {𝐴}) = 𝑇)
22	20, 21	sylib 220	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝑇 ∪ {𝐴}) = 𝑇)
23	18, 22	eqtrid 2809	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = 𝑇)
24	23	unieqd 4878	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = ∪ 𝑇)
25	24, 5	eqtrd 2797	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (^◡𝐹 “ 𝑈))
26	17, 25	eqtr3id 2811	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∪ ∪ {𝐴}) = (^◡𝐹 “ 𝑈))
27	16, 26	eqtr3d 2799	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∪ 𝐴) = (^◡𝐹 “ 𝑈))
28		difss 4089	. . . . . . 7 ⊢ (𝑇 ∖ {𝐴}) ⊆ 𝑇
29	28	unissi 4874	. . . . . 6 ⊢ ∪ (𝑇 ∖ {𝐴}) ⊆ ∪ 𝑇
30	29, 5	sseqtrid 3978	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ (𝑇 ∖ {𝐴}) ⊆ (^◡𝐹 “ 𝑈))
31		uniiun 5016	. . . . . . . 8 ⊢ ∪ (𝑇 ∖ {𝐴}) = ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥
32	31	ineq2i 4169	. . . . . . 7 ⊢ (𝐴 ∩ ∪ (𝑇 ∖ {𝐴})) = (𝐴 ∩ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥)
33		incom 4161	. . . . . . 7 ⊢ (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = (𝐴 ∩ ∪ (𝑇 ∖ {𝐴}))
34		iunin2 5028	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥)
35	32, 33, 34	3eqtr4i 2795	. . . . . 6 ⊢ (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥)
36		eldifsn 4746	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑇 ∖ {𝐴}) ↔ (𝑥 ∈ 𝑇 ∧ 𝑥 ≠ 𝐴))
37		nesym 3013	. . . . . . . . . . . 12 ⊢ (𝑥 ≠ 𝐴 ↔ ¬ 𝐴 = 𝑥)
38	3	cvmsdisj 35620	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇 ∧ 𝑥 ∈ 𝑇) → (𝐴 = 𝑥 ∨ (𝐴 ∩ 𝑥) = ∅))
39	38	3expa 1131	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → (𝐴 = 𝑥 ∨ (𝐴 ∩ 𝑥) = ∅))
40	39	ord 875	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → (¬ 𝐴 = 𝑥 → (𝐴 ∩ 𝑥) = ∅))
41	37, 40	biimtrid 244	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → (𝑥 ≠ 𝐴 → (𝐴 ∩ 𝑥) = ∅))
42	41	impr 458	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ (𝑥 ∈ 𝑇 ∧ 𝑥 ≠ 𝐴)) → (𝐴 ∩ 𝑥) = ∅)
43	36, 42	sylan2b 603	. . . . . . . . 9 ⊢ (((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ (𝑇 ∖ {𝐴})) → (𝐴 ∩ 𝑥) = ∅)
44	43	iuneq2dv 4974	. . . . . . . 8 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥) = ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})∅)
45	44	3adant1 1143	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥) = ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})∅)
46		iun0 5019	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})∅ = ∅
47	45, 46	eqtrdi 2813	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥) = ∅)
48	35, 47	eqtrid 2809	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∅)
49		uneqdifeq 4446	. . . . 5 ⊢ ((∪ (𝑇 ∖ {𝐴}) ⊆ (^◡𝐹 “ 𝑈) ∧ (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∅) → ((∪ (𝑇 ∖ {𝐴}) ∪ 𝐴) = (^◡𝐹 “ 𝑈) ↔ ((^◡𝐹 “ 𝑈) ∖ ∪ (𝑇 ∖ {𝐴})) = 𝐴))
50	30, 48, 49	syl2anc 593	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ((∪ (𝑇 ∖ {𝐴}) ∪ 𝐴) = (^◡𝐹 “ 𝑈) ↔ ((^◡𝐹 “ 𝑈) ∖ ∪ (𝑇 ∖ {𝐴})) = 𝐴))
51	27, 50	mpbid 234	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ((^◡𝐹 “ 𝑈) ∖ ∪ (𝑇 ∖ {𝐴})) = 𝐴)
52	13, 51	eqtr3d 2799	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝐶 ↾_t (^◡𝐹 “ 𝑈)) ∖ ∪ (𝑇 ∖ {𝐴})) = 𝐴)
53		uniexg 7723	. . . . . 6 ⊢ (𝑇 ∈ (𝑆‘𝑈) → ∪ 𝑇 ∈ V)
54	53	3ad2ant2 1147	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑇 ∈ V)
55	5, 54	eqeltrrd 2863	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (^◡𝐹 “ 𝑈) ∈ V)
56		resttop 23220	. . . 4 ⊢ ((𝐶 ∈ Top ∧ (^◡𝐹 “ 𝑈) ∈ V) → (𝐶 ↾_t (^◡𝐹 “ 𝑈)) ∈ Top)
57	2, 55, 56	syl2anc 593	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐶 ↾_t (^◡𝐹 “ 𝑈)) ∈ Top)
58		elssuni 4897	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑇 → 𝑥 ⊆ ∪ 𝑇)
59	58	adantl 485	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝑥 ⊆ ∪ 𝑇)
60	5	adantr 484	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → ∪ 𝑇 = (^◡𝐹 “ 𝑈))
61	59, 60	sseqtrd 3972	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝑥 ⊆ (^◡𝐹 “ 𝑈))
62		dfss2 3922	. . . . . . . . 9 ⊢ (𝑥 ⊆ (^◡𝐹 “ 𝑈) ↔ (𝑥 ∩ (^◡𝐹 “ 𝑈)) = 𝑥)
63	61, 62	sylib 220	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → (𝑥 ∩ (^◡𝐹 “ 𝑈)) = 𝑥)
64	2	adantr 484	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝐶 ∈ Top)
65	55	adantr 484	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → (^◡𝐹 “ 𝑈) ∈ V)
66	7	sselda 3936	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝐶)
67		elrestr 17457	. . . . . . . . 9 ⊢ ((𝐶 ∈ Top ∧ (^◡𝐹 “ 𝑈) ∈ V ∧ 𝑥 ∈ 𝐶) → (𝑥 ∩ (^◡𝐹 “ 𝑈)) ∈ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
68	64, 65, 66, 67	syl3anc 1390	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → (𝑥 ∩ (^◡𝐹 “ 𝑈)) ∈ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
69	63, 68	eqeltrrd 2863	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
70	69	ex 416	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝑥 ∈ 𝑇 → 𝑥 ∈ (𝐶 ↾_t (^◡𝐹 “ 𝑈))))
71	70	ssrdv 3942	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝑇 ⊆ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
72	71	ssdifssd 4100	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝑇 ∖ {𝐴}) ⊆ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
73		uniopn 22957	. . . 4 ⊢ (((𝐶 ↾_t (^◡𝐹 “ 𝑈)) ∈ Top ∧ (𝑇 ∖ {𝐴}) ⊆ (𝐶 ↾_t (^◡𝐹 “ 𝑈))) → ∪ (𝑇 ∖ {𝐴}) ∈ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
74	57, 72, 73	syl2anc 593	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ (𝑇 ∖ {𝐴}) ∈ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
75		eqid 2762	. . . 4 ⊢ ∪ (𝐶 ↾_t (^◡𝐹 “ 𝑈)) = ∪ (𝐶 ↾_t (^◡𝐹 “ 𝑈))
76	75	opncld 23093	. . 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 593	. 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 2863	1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ (Clsd‘(𝐶 ↾_t (^◡𝐹 “ 𝑈))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 ∀wral 3076 {crab 3414 Vcvv 3454 ∖ cdif 3901 ∪ cun 3902 ∩ cin 3903 ⊆ wss 3904 ∅c0 4285 𝒫 cpw 4555 {csn 4582 ∪ cuni 4865 ∪ ciun 4949 ↦ cmpt 5181 ^◡ccnv 5646 ↾ cres 5649 “ cima 5650 ‘cfv 6521 (class class class)co 7396 ↾_t crest 17449 Topctop 22953 Clsdccld 23076 Homeochmeo 23813 CovMap ccvm 35605

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3077 df-rex 3087 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-en 8928 df-fin 8931 df-fi 9357 df-rest 17451 df-topgen 17472 df-top 22954 df-topon 22971 df-bases 23006 df-cld 23079 df-cvm 35606

This theorem is referenced by: cvmliftmolem1 35631 cvmlift2lem9 35661 cvmlift3lem6 35674

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