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Theorem cvmscld 31562
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 31549 . . . . . 6 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
213ad2ant1 1128 . . . . 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 31558 . . . . . . 7 (𝑇 ∈ (𝑆𝑈) → 𝑇 = (𝐹𝑈))
543ad2ant2 1129 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇 = (𝐹𝑈))
63cvmsss 31556 . . . . . . . 8 (𝑇 ∈ (𝑆𝑈) → 𝑇𝐶)
763ad2ant2 1129 . . . . . . 7 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇𝐶)
87unissd 4614 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇 𝐶)
95, 8eqsstr3d 3781 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝑈) ⊆ 𝐶)
10 eqid 2760 . . . . . 6 𝐶 = 𝐶
1110restuni 21168 . . . . 5 ((𝐶 ∈ Top ∧ (𝐹𝑈) ⊆ 𝐶) → (𝐹𝑈) = (𝐶t (𝐹𝑈)))
122, 9, 11syl2anc 696 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝑈) = (𝐶t (𝐹𝑈)))
1312difeq1d 3870 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝐹𝑈) ∖ (𝑇 ∖ {𝐴})) = ( (𝐶t (𝐹𝑈)) ∖ (𝑇 ∖ {𝐴})))
14 unisng 4604 . . . . . . 7 (𝐴𝑇 {𝐴} = 𝐴)
15143ad2ant3 1130 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → {𝐴} = 𝐴)
1615uneq2d 3910 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝑇 ∖ {𝐴}) ∪ {𝐴}) = ( (𝑇 ∖ {𝐴}) ∪ 𝐴))
17 uniun 4608 . . . . . 6 ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = ( (𝑇 ∖ {𝐴}) ∪ {𝐴})
18 undif1 4187 . . . . . . . . 9 ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (𝑇 ∪ {𝐴})
19 simp3 1133 . . . . . . . . . . 11 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴𝑇)
2019snssd 4485 . . . . . . . . . 10 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → {𝐴} ⊆ 𝑇)
21 ssequn2 3929 . . . . . . . . . 10 ({𝐴} ⊆ 𝑇 ↔ (𝑇 ∪ {𝐴}) = 𝑇)
2220, 21sylib 208 . . . . . . . . 9 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑇 ∪ {𝐴}) = 𝑇)
2318, 22syl5eq 2806 . . . . . . . 8 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = 𝑇)
2423unieqd 4598 . . . . . . 7 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = 𝑇)
2524, 5eqtrd 2794 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (𝐹𝑈))
2617, 25syl5eqr 2808 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝑇 ∖ {𝐴}) ∪ {𝐴}) = (𝐹𝑈))
2716, 26eqtr3d 2796 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝑇 ∖ {𝐴}) ∪ 𝐴) = (𝐹𝑈))
28 difss 3880 . . . . . . 7 (𝑇 ∖ {𝐴}) ⊆ 𝑇
2928unissi 4613 . . . . . 6 (𝑇 ∖ {𝐴}) ⊆ 𝑇
3029, 5syl5sseq 3794 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑇 ∖ {𝐴}) ⊆ (𝐹𝑈))
31 uniiun 4725 . . . . . . . 8 (𝑇 ∖ {𝐴}) = 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥
3231ineq2i 3954 . . . . . . 7 (𝐴 (𝑇 ∖ {𝐴})) = (𝐴 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥)
33 incom 3948 . . . . . . 7 ( (𝑇 ∖ {𝐴}) ∩ 𝐴) = (𝐴 (𝑇 ∖ {𝐴}))
34 iunin2 4736 . . . . . . 7 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥) = (𝐴 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥)
3532, 33, 343eqtr4i 2792 . . . . . 6 ( (𝑇 ∖ {𝐴}) ∩ 𝐴) = 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥)
36 eldifsn 4462 . . . . . . . . . 10 (𝑥 ∈ (𝑇 ∖ {𝐴}) ↔ (𝑥𝑇𝑥𝐴))
37 nesym 2988 . . . . . . . . . . . 12 (𝑥𝐴 ↔ ¬ 𝐴 = 𝑥)
383cvmsdisj 31559 . . . . . . . . . . . . . 14 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝑥𝑇) → (𝐴 = 𝑥 ∨ (𝐴𝑥) = ∅))
39383expa 1112 . . . . . . . . . . . . 13 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (𝐴 = 𝑥 ∨ (𝐴𝑥) = ∅))
4039ord 391 . . . . . . . . . . . 12 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (¬ 𝐴 = 𝑥 → (𝐴𝑥) = ∅))
4137, 40syl5bi 232 . . . . . . . . . . 11 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (𝑥𝐴 → (𝐴𝑥) = ∅))
4241impr 650 . . . . . . . . . 10 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ (𝑥𝑇𝑥𝐴)) → (𝐴𝑥) = ∅)
4336, 42sylan2b 493 . . . . . . . . 9 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥 ∈ (𝑇 ∖ {𝐴})) → (𝐴𝑥) = ∅)
4443iuneq2dv 4694 . . . . . . . 8 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥) = 𝑥 ∈ (𝑇 ∖ {𝐴})∅)
45443adant1 1125 . . . . . . 7 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥) = 𝑥 ∈ (𝑇 ∖ {𝐴})∅)
46 iun0 4728 . . . . . . 7 𝑥 ∈ (𝑇 ∖ {𝐴})∅ = ∅
4745, 46syl6eq 2810 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥) = ∅)
4835, 47syl5eq 2806 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∅)
49 uneqdifeq 4201 . . . . 5 (( (𝑇 ∖ {𝐴}) ⊆ (𝐹𝑈) ∧ ( (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∅) → (( (𝑇 ∖ {𝐴}) ∪ 𝐴) = (𝐹𝑈) ↔ ((𝐹𝑈) ∖ (𝑇 ∖ {𝐴})) = 𝐴))
5030, 48, 49syl2anc 696 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (( (𝑇 ∖ {𝐴}) ∪ 𝐴) = (𝐹𝑈) ↔ ((𝐹𝑈) ∖ (𝑇 ∖ {𝐴})) = 𝐴))
5127, 50mpbid 222 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝐹𝑈) ∖ (𝑇 ∖ {𝐴})) = 𝐴)
5213, 51eqtr3d 2796 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝐶t (𝐹𝑈)) ∖ (𝑇 ∖ {𝐴})) = 𝐴)
53 uniexg 7120 . . . . . 6 (𝑇 ∈ (𝑆𝑈) → 𝑇 ∈ V)
54533ad2ant2 1129 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇 ∈ V)
555, 54eqeltrrd 2840 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝑈) ∈ V)
56 resttop 21166 . . . 4 ((𝐶 ∈ Top ∧ (𝐹𝑈) ∈ V) → (𝐶t (𝐹𝑈)) ∈ Top)
572, 55, 56syl2anc 696 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐶t (𝐹𝑈)) ∈ Top)
58 elssuni 4619 . . . . . . . . . . 11 (𝑥𝑇𝑥 𝑇)
5958adantl 473 . . . . . . . . . 10 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝑥 𝑇)
605adantr 472 . . . . . . . . . 10 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝑇 = (𝐹𝑈))
6159, 60sseqtrd 3782 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝑥 ⊆ (𝐹𝑈))
62 df-ss 3729 . . . . . . . . 9 (𝑥 ⊆ (𝐹𝑈) ↔ (𝑥 ∩ (𝐹𝑈)) = 𝑥)
6361, 62sylib 208 . . . . . . . 8 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (𝑥 ∩ (𝐹𝑈)) = 𝑥)
642adantr 472 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝐶 ∈ Top)
6555adantr 472 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (𝐹𝑈) ∈ V)
667sselda 3744 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝑥𝐶)
67 elrestr 16291 . . . . . . . . 9 ((𝐶 ∈ Top ∧ (𝐹𝑈) ∈ V ∧ 𝑥𝐶) → (𝑥 ∩ (𝐹𝑈)) ∈ (𝐶t (𝐹𝑈)))
6864, 65, 66, 67syl3anc 1477 . . . . . . . 8 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (𝑥 ∩ (𝐹𝑈)) ∈ (𝐶t (𝐹𝑈)))
6963, 68eqeltrrd 2840 . . . . . . 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 3750 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇 ⊆ (𝐶t (𝐹𝑈)))
7271ssdifssd 3891 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑇 ∖ {𝐴}) ⊆ (𝐶t (𝐹𝑈)))
73 uniopn 20904 . . . 4 (((𝐶t (𝐹𝑈)) ∈ Top ∧ (𝑇 ∖ {𝐴}) ⊆ (𝐶t (𝐹𝑈))) → (𝑇 ∖ {𝐴}) ∈ (𝐶t (𝐹𝑈)))
7457, 72, 73syl2anc 696 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑇 ∖ {𝐴}) ∈ (𝐶t (𝐹𝑈)))
75 eqid 2760 . . . 4 (𝐶t (𝐹𝑈)) = (𝐶t (𝐹𝑈))
7675opncld 21039 . . 3 (((𝐶t (𝐹𝑈)) ∈ Top ∧ (𝑇 ∖ {𝐴}) ∈ (𝐶t (𝐹𝑈))) → ( (𝐶t (𝐹𝑈)) ∖ (𝑇 ∖ {𝐴})) ∈ (Clsd‘(𝐶t (𝐹𝑈))))
7757, 74, 76syl2anc 696 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝐶t (𝐹𝑈)) ∖ (𝑇 ∖ {𝐴})) ∈ (Clsd‘(𝐶t (𝐹𝑈))))
7852, 77eqeltrrd 2840 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 196  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  {crab 3054  Vcvv 3340  cdif 3712  cun 3713  cin 3714  wss 3715  c0 4058  𝒫 cpw 4302  {csn 4321   cuni 4588   ciun 4672  cmpt 4881  ccnv 5265  cres 5268  cima 5269  cfv 6049  (class class class)co 6813  t crest 16283  Topctop 20900  Clsdccld 21022  Homeochmeo 21758   CovMap ccvm 31544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-oadd 7733  df-er 7911  df-en 8122  df-fin 8125  df-fi 8482  df-rest 16285  df-topgen 16306  df-top 20901  df-topon 20918  df-bases 20952  df-cld 21025  df-cvm 31545
This theorem is referenced by:  cvmliftmolem1  31570  cvmlift2lem9  31600  cvmlift3lem6  31613
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