cvmscld - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmscld		Structured version Visualization version GIF version

Theorem cvmscld 32633

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 32620	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
2	1	3ad2ant1 1130	. . . . 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 32629	. . . . . . 7 ⊢ (𝑇 ∈ (𝑆‘𝑈) → ∪ 𝑇 = (^◡𝐹 “ 𝑈))
5	4	3ad2ant2 1131	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑇 = (^◡𝐹 “ 𝑈))
6	3	cvmsss 32627	. . . . . . . 8 ⊢ (𝑇 ∈ (𝑆‘𝑈) → 𝑇 ⊆ 𝐶)
7	6	3ad2ant2 1131	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝑇 ⊆ 𝐶)
8	7	unissd 4810	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑇 ⊆ ∪ 𝐶)
9	5, 8	eqsstrrd 3954	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (^◡𝐹 “ 𝑈) ⊆ ∪ 𝐶)
10		eqid 2798	. . . . . 6 ⊢ ∪ 𝐶 = ∪ 𝐶
11	10	restuni 21767	. . . . 5 ⊢ ((𝐶 ∈ Top ∧ (^◡𝐹 “ 𝑈) ⊆ ∪ 𝐶) → (^◡𝐹 “ 𝑈) = ∪ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
12	2, 9, 11	syl2anc 587	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (^◡𝐹 “ 𝑈) = ∪ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
13	12	difeq1d 4049	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ((^◡𝐹 “ 𝑈) ∖ ∪ (𝑇 ∖ {𝐴})) = (∪ (𝐶 ↾_t (^◡𝐹 “ 𝑈)) ∖ ∪ (𝑇 ∖ {𝐴})))
14		unisng 4819	. . . . . . 7 ⊢ (𝐴 ∈ 𝑇 → ∪ {𝐴} = 𝐴)
15	14	3ad2ant3 1132	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ {𝐴} = 𝐴)
16	15	uneq2d 4090	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∪ ∪ {𝐴}) = (∪ (𝑇 ∖ {𝐴}) ∪ 𝐴))
17		uniun 4823	. . . . . 6 ⊢ ∪ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (∪ (𝑇 ∖ {𝐴}) ∪ ∪ {𝐴})
18		undif1 4382	. . . . . . . . 9 ⊢ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (𝑇 ∪ {𝐴})
19		simp3 1135	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝐴 ∈ 𝑇)
20	19	snssd 4702	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → {𝐴} ⊆ 𝑇)
21		ssequn2 4110	. . . . . . . . . 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	syl5eq 2845	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = 𝑇)
24	23	unieqd 4814	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = ∪ 𝑇)
25	24, 5	eqtrd 2833	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (^◡𝐹 “ 𝑈))
26	17, 25	syl5eqr 2847	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∪ ∪ {𝐴}) = (^◡𝐹 “ 𝑈))
27	16, 26	eqtr3d 2835	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∪ 𝐴) = (^◡𝐹 “ 𝑈))
28		difss 4059	. . . . . . 7 ⊢ (𝑇 ∖ {𝐴}) ⊆ 𝑇
29	28	unissi 4809	. . . . . 6 ⊢ ∪ (𝑇 ∖ {𝐴}) ⊆ ∪ 𝑇
30	29, 5	sseqtrid 3967	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ (𝑇 ∖ {𝐴}) ⊆ (^◡𝐹 “ 𝑈))
31		uniiun 4945	. . . . . . . 8 ⊢ ∪ (𝑇 ∖ {𝐴}) = ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥
32	31	ineq2i 4136	. . . . . . 7 ⊢ (𝐴 ∩ ∪ (𝑇 ∖ {𝐴})) = (𝐴 ∩ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥)
33		incom 4128	. . . . . . 7 ⊢ (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = (𝐴 ∩ ∪ (𝑇 ∖ {𝐴}))
34		iunin2 4956	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥)
35	32, 33, 34	3eqtr4i 2831	. . . . . 6 ⊢ (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥)
36		eldifsn 4680	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑇 ∖ {𝐴}) ↔ (𝑥 ∈ 𝑇 ∧ 𝑥 ≠ 𝐴))
37		nesym 3043	. . . . . . . . . . . 12 ⊢ (𝑥 ≠ 𝐴 ↔ ¬ 𝐴 = 𝑥)
38	3	cvmsdisj 32630	. . . . . . . . . . . . . 14 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇 ∧ 𝑥 ∈ 𝑇) → (𝐴 = 𝑥 ∨ (𝐴 ∩ 𝑥) = ∅))
39	38	3expa 1115	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → (𝐴 = 𝑥 ∨ (𝐴 ∩ 𝑥) = ∅))
40	39	ord 861	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → (¬ 𝐴 = 𝑥 → (𝐴 ∩ 𝑥) = ∅))
41	37, 40	syl5bi 245	. . . . . . . . . . 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 596	. . . . . . . . 9 ⊢ (((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ (𝑇 ∖ {𝐴})) → (𝐴 ∩ 𝑥) = ∅)
44	43	iuneq2dv 4905	. . . . . . . 8 ⊢ ((𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥) = ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})∅)
45	44	3adant1 1127	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥) = ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})∅)
46		iun0 4948	. . . . . . 7 ⊢ ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})∅ = ∅
47	45, 46	eqtrdi 2849	. . . . . 6 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴 ∩ 𝑥) = ∅)
48	35, 47	syl5eq 2845	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∅)
49		uneqdifeq 4396	. . . . 5 ⊢ ((∪ (𝑇 ∖ {𝐴}) ⊆ (^◡𝐹 “ 𝑈) ∧ (∪ (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∅) → ((∪ (𝑇 ∖ {𝐴}) ∪ 𝐴) = (^◡𝐹 “ 𝑈) ↔ ((^◡𝐹 “ 𝑈) ∖ ∪ (𝑇 ∖ {𝐴})) = 𝐴))
50	30, 48, 49	syl2anc 587	. . . 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 2835	. 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (∪ (𝐶 ↾_t (^◡𝐹 “ 𝑈)) ∖ ∪ (𝑇 ∖ {𝐴})) = 𝐴)
53		uniexg 7446	. . . . . 6 ⊢ (𝑇 ∈ (𝑆‘𝑈) → ∪ 𝑇 ∈ V)
54	53	3ad2ant2 1131	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ 𝑇 ∈ V)
55	5, 54	eqeltrrd 2891	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (^◡𝐹 “ 𝑈) ∈ V)
56		resttop 21765	. . . 4 ⊢ ((𝐶 ∈ Top ∧ (^◡𝐹 “ 𝑈) ∈ V) → (𝐶 ↾_t (^◡𝐹 “ 𝑈)) ∈ Top)
57	2, 55, 56	syl2anc 587	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝐶 ↾_t (^◡𝐹 “ 𝑈)) ∈ Top)
58		elssuni 4830	. . . . . . . . . . 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 3955	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝑥 ⊆ (^◡𝐹 “ 𝑈))
62		df-ss 3898	. . . . . . . . 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 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 3915	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝐶)
67		elrestr 16694	. . . . . . . . 9 ⊢ ((𝐶 ∈ Top ∧ (^◡𝐹 “ 𝑈) ∈ V ∧ 𝑥 ∈ 𝐶) → (𝑥 ∩ (^◡𝐹 “ 𝑈)) ∈ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
68	64, 65, 66, 67	syl3anc 1368	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) ∧ 𝑥 ∈ 𝑇) → (𝑥 ∩ (^◡𝐹 “ 𝑈)) ∈ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
69	63, 68	eqeltrrd 2891	. . . . . . 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 3921	. . . . 5 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → 𝑇 ⊆ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
72	71	ssdifssd 4070	. . . 4 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → (𝑇 ∖ {𝐴}) ⊆ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
73		uniopn 21502	. . . 4 ⊢ (((𝐶 ↾_t (^◡𝐹 “ 𝑈)) ∈ Top ∧ (𝑇 ∖ {𝐴}) ⊆ (𝐶 ↾_t (^◡𝐹 “ 𝑈))) → ∪ (𝑇 ∖ {𝐴}) ∈ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
74	57, 72, 73	syl2anc 587	. . 3 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆‘𝑈) ∧ 𝐴 ∈ 𝑇) → ∪ (𝑇 ∖ {𝐴}) ∈ (𝐶 ↾_t (^◡𝐹 “ 𝑈)))
75		eqid 2798	. . . 4 ⊢ ∪ (𝐶 ↾_t (^◡𝐹 “ 𝑈)) = ∪ (𝐶 ↾_t (^◡𝐹 “ 𝑈))
76	75	opncld 21638	. . 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 587	. 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 2891	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 209 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 {crab 3110 Vcvv 3441 ∖ cdif 3878 ∪ cun 3879 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 {csn 4525 ∪ cuni 4800 ∪ ciun 4881 ↦ cmpt 5110 ^◡ccnv 5518 ↾ cres 5521 “ cima 5522 ‘cfv 6324 (class class class)co 7135 ↾_t crest 16686 Topctop 21498 Clsdccld 21621 Homeochmeo 22358 CovMap ccvm 32615

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-oadd 8089 df-er 8272 df-en 8493 df-fin 8496 df-fi 8859 df-rest 16688 df-topgen 16709 df-top 21499 df-topon 21516 df-bases 21551 df-cld 21624 df-cvm 32616

This theorem is referenced by: cvmliftmolem1 32641 cvmlift2lem9 32671 cvmlift3lem6 32684

W3C validator