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Theorem cvmscld 30509
 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 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
Assertion
Ref Expression
cvmscld ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴 ∈ (Clsd‘(𝐶t (𝐹𝑈))))
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.
StepHypRef Expression
1 cvmtop1 30496 . . . . . 6 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
213ad2ant1 1075 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐶 ∈ Top)
3 cvmcov.1 . . . . . . . 8 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
43cvmsuni 30505 . . . . . . 7 (𝑇 ∈ (𝑆𝑈) → 𝑇 = (𝐹𝑈))
543ad2ant2 1076 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇 = (𝐹𝑈))
63cvmsss 30503 . . . . . . . 8 (𝑇 ∈ (𝑆𝑈) → 𝑇𝐶)
763ad2ant2 1076 . . . . . . 7 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇𝐶)
87unissd 4398 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇 𝐶)
95, 8eqsstr3d 3603 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝑈) ⊆ 𝐶)
10 eqid 2610 . . . . . 6 𝐶 = 𝐶
1110restuni 20776 . . . . 5 ((𝐶 ∈ Top ∧ (𝐹𝑈) ⊆ 𝐶) → (𝐹𝑈) = (𝐶t (𝐹𝑈)))
122, 9, 11syl2anc 691 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝑈) = (𝐶t (𝐹𝑈)))
1312difeq1d 3689 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝐹𝑈) ∖ (𝑇 ∖ {𝐴})) = ( (𝐶t (𝐹𝑈)) ∖ (𝑇 ∖ {𝐴})))
14 unisng 4388 . . . . . . 7 (𝐴𝑇 {𝐴} = 𝐴)
15143ad2ant3 1077 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → {𝐴} = 𝐴)
1615uneq2d 3729 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝑇 ∖ {𝐴}) ∪ {𝐴}) = ( (𝑇 ∖ {𝐴}) ∪ 𝐴))
17 uniun 4392 . . . . . 6 ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = ( (𝑇 ∖ {𝐴}) ∪ {𝐴})
18 undif1 3995 . . . . . . . . 9 ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (𝑇 ∪ {𝐴})
19 simp3 1056 . . . . . . . . . . 11 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴𝑇)
2019snssd 4281 . . . . . . . . . 10 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → {𝐴} ⊆ 𝑇)
21 ssequn2 3748 . . . . . . . . . 10 ({𝐴} ⊆ 𝑇 ↔ (𝑇 ∪ {𝐴}) = 𝑇)
2220, 21sylib 207 . . . . . . . . 9 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑇 ∪ {𝐴}) = 𝑇)
2318, 22syl5eq 2656 . . . . . . . 8 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = 𝑇)
2423unieqd 4382 . . . . . . 7 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = 𝑇)
2524, 5eqtrd 2644 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (𝐹𝑈))
2617, 25syl5eqr 2658 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝑇 ∖ {𝐴}) ∪ {𝐴}) = (𝐹𝑈))
2716, 26eqtr3d 2646 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝑇 ∖ {𝐴}) ∪ 𝐴) = (𝐹𝑈))
28 difss 3699 . . . . . . 7 (𝑇 ∖ {𝐴}) ⊆ 𝑇
2928unissi 4397 . . . . . 6 (𝑇 ∖ {𝐴}) ⊆ 𝑇
3029, 5syl5sseq 3616 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑇 ∖ {𝐴}) ⊆ (𝐹𝑈))
31 uniiun 4509 . . . . . . . 8 (𝑇 ∖ {𝐴}) = 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥
3231ineq2i 3773 . . . . . . 7 (𝐴 (𝑇 ∖ {𝐴})) = (𝐴 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥)
33 incom 3767 . . . . . . 7 ( (𝑇 ∖ {𝐴}) ∩ 𝐴) = (𝐴 (𝑇 ∖ {𝐴}))
34 iunin2 4520 . . . . . . 7 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥) = (𝐴 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥)
3532, 33, 343eqtr4i 2642 . . . . . 6 ( (𝑇 ∖ {𝐴}) ∩ 𝐴) = 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥)
36 eldifsn 4260 . . . . . . . . . 10 (𝑥 ∈ (𝑇 ∖ {𝐴}) ↔ (𝑥𝑇𝑥𝐴))
37 nesym 2838 . . . . . . . . . . . 12 (𝑥𝐴 ↔ ¬ 𝐴 = 𝑥)
383cvmsdisj 30506 . . . . . . . . . . . . . 14 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝑥𝑇) → (𝐴 = 𝑥 ∨ (𝐴𝑥) = ∅))
39383expa 1257 . . . . . . . . . . . . 13 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (𝐴 = 𝑥 ∨ (𝐴𝑥) = ∅))
4039ord 391 . . . . . . . . . . . 12 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (¬ 𝐴 = 𝑥 → (𝐴𝑥) = ∅))
4137, 40syl5bi 231 . . . . . . . . . . 11 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (𝑥𝐴 → (𝐴𝑥) = ∅))
4241impr 647 . . . . . . . . . 10 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ (𝑥𝑇𝑥𝐴)) → (𝐴𝑥) = ∅)
4336, 42sylan2b 491 . . . . . . . . 9 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥 ∈ (𝑇 ∖ {𝐴})) → (𝐴𝑥) = ∅)
4443iuneq2dv 4478 . . . . . . . 8 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥) = 𝑥 ∈ (𝑇 ∖ {𝐴})∅)
45443adant1 1072 . . . . . . 7 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥) = 𝑥 ∈ (𝑇 ∖ {𝐴})∅)
46 iun0 4512 . . . . . . 7 𝑥 ∈ (𝑇 ∖ {𝐴})∅ = ∅
4745, 46syl6eq 2660 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥) = ∅)
4835, 47syl5eq 2656 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∅)
49 uneqdifeq 4009 . . . . 5 (( (𝑇 ∖ {𝐴}) ⊆ (𝐹𝑈) ∧ ( (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∅) → (( (𝑇 ∖ {𝐴}) ∪ 𝐴) = (𝐹𝑈) ↔ ((𝐹𝑈) ∖ (𝑇 ∖ {𝐴})) = 𝐴))
5030, 48, 49syl2anc 691 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (( (𝑇 ∖ {𝐴}) ∪ 𝐴) = (𝐹𝑈) ↔ ((𝐹𝑈) ∖ (𝑇 ∖ {𝐴})) = 𝐴))
5127, 50mpbid 221 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝐹𝑈) ∖ (𝑇 ∖ {𝐴})) = 𝐴)
5213, 51eqtr3d 2646 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝐶t (𝐹𝑈)) ∖ (𝑇 ∖ {𝐴})) = 𝐴)
53 uniexg 6853 . . . . . 6 (𝑇 ∈ (𝑆𝑈) → 𝑇 ∈ V)
54533ad2ant2 1076 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇 ∈ V)
555, 54eqeltrrd 2689 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝑈) ∈ V)
56 resttop 20774 . . . 4 ((𝐶 ∈ Top ∧ (𝐹𝑈) ∈ V) → (𝐶t (𝐹𝑈)) ∈ Top)
572, 55, 56syl2anc 691 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐶t (𝐹𝑈)) ∈ Top)
58 elssuni 4403 . . . . . . . . . . 11 (𝑥𝑇𝑥 𝑇)
5958adantl 481 . . . . . . . . . 10 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝑥 𝑇)
605adantr 480 . . . . . . . . . 10 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝑇 = (𝐹𝑈))
6159, 60sseqtrd 3604 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝑥 ⊆ (𝐹𝑈))
62 df-ss 3554 . . . . . . . . 9 (𝑥 ⊆ (𝐹𝑈) ↔ (𝑥 ∩ (𝐹𝑈)) = 𝑥)
6361, 62sylib 207 . . . . . . . 8 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (𝑥 ∩ (𝐹𝑈)) = 𝑥)
642adantr 480 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝐶 ∈ Top)
6555adantr 480 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (𝐹𝑈) ∈ V)
667sselda 3568 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝑥𝐶)
67 elrestr 15912 . . . . . . . . 9 ((𝐶 ∈ Top ∧ (𝐹𝑈) ∈ V ∧ 𝑥𝐶) → (𝑥 ∩ (𝐹𝑈)) ∈ (𝐶t (𝐹𝑈)))
6864, 65, 66, 67syl3anc 1318 . . . . . . . 8 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (𝑥 ∩ (𝐹𝑈)) ∈ (𝐶t (𝐹𝑈)))
6963, 68eqeltrrd 2689 . . . . . . 7 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝑥 ∈ (𝐶t (𝐹𝑈)))
7069ex 449 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑥𝑇𝑥 ∈ (𝐶t (𝐹𝑈))))
7170ssrdv 3574 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇 ⊆ (𝐶t (𝐹𝑈)))
7271ssdifssd 3710 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑇 ∖ {𝐴}) ⊆ (𝐶t (𝐹𝑈)))
73 uniopn 20527 . . . 4 (((𝐶t (𝐹𝑈)) ∈ Top ∧ (𝑇 ∖ {𝐴}) ⊆ (𝐶t (𝐹𝑈))) → (𝑇 ∖ {𝐴}) ∈ (𝐶t (𝐹𝑈)))
7457, 72, 73syl2anc 691 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑇 ∖ {𝐴}) ∈ (𝐶t (𝐹𝑈)))
75 eqid 2610 . . . 4 (𝐶t (𝐹𝑈)) = (𝐶t (𝐹𝑈))
7675opncld 20647 . . 3 (((𝐶t (𝐹𝑈)) ∈ Top ∧ (𝑇 ∖ {𝐴}) ∈ (𝐶t (𝐹𝑈))) → ( (𝐶t (𝐹𝑈)) ∖ (𝑇 ∖ {𝐴})) ∈ (Clsd‘(𝐶t (𝐹𝑈))))
7757, 74, 76syl2anc 691 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝐶t (𝐹𝑈)) ∖ (𝑇 ∖ {𝐴})) ∈ (Clsd‘(𝐶t (𝐹𝑈))))
7852, 77eqeltrrd 2689 1 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴 ∈ (Clsd‘(𝐶t (𝐹𝑈))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  ∪ ciun 4455   ↦ cmpt 4643  ◡ccnv 5037   ↾ cres 5040   “ cima 5041  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904  Topctop 20517  Clsdccld 20630  Homeochmeo 21366   CovMap ccvm 30491 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-cvm 30492 This theorem is referenced by:  cvmliftmolem1  30517  cvmlift2lem9  30547  cvmlift3lem6  30560
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