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Theorem cvmscld 31706
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 31693 . . . . . 6 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
213ad2ant1 1163 . . . . 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 31702 . . . . . . 7 (𝑇 ∈ (𝑆𝑈) → 𝑇 = (𝐹𝑈))
543ad2ant2 1164 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇 = (𝐹𝑈))
63cvmsss 31700 . . . . . . . 8 (𝑇 ∈ (𝑆𝑈) → 𝑇𝐶)
763ad2ant2 1164 . . . . . . 7 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇𝐶)
87unissd 4622 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇 𝐶)
95, 8eqsstr3d 3802 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝑈) ⊆ 𝐶)
10 eqid 2765 . . . . . 6 𝐶 = 𝐶
1110restuni 21249 . . . . 5 ((𝐶 ∈ Top ∧ (𝐹𝑈) ⊆ 𝐶) → (𝐹𝑈) = (𝐶t (𝐹𝑈)))
122, 9, 11syl2anc 579 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝑈) = (𝐶t (𝐹𝑈)))
1312difeq1d 3891 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝐹𝑈) ∖ (𝑇 ∖ {𝐴})) = ( (𝐶t (𝐹𝑈)) ∖ (𝑇 ∖ {𝐴})))
14 unisng 4611 . . . . . . 7 (𝐴𝑇 {𝐴} = 𝐴)
15143ad2ant3 1165 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → {𝐴} = 𝐴)
1615uneq2d 3931 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝑇 ∖ {𝐴}) ∪ {𝐴}) = ( (𝑇 ∖ {𝐴}) ∪ 𝐴))
17 uniun 4617 . . . . . 6 ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = ( (𝑇 ∖ {𝐴}) ∪ {𝐴})
18 undif1 4205 . . . . . . . . 9 ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (𝑇 ∪ {𝐴})
19 simp3 1168 . . . . . . . . . . 11 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴𝑇)
2019snssd 4496 . . . . . . . . . 10 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → {𝐴} ⊆ 𝑇)
21 ssequn2 3950 . . . . . . . . . 10 ({𝐴} ⊆ 𝑇 ↔ (𝑇 ∪ {𝐴}) = 𝑇)
2220, 21sylib 209 . . . . . . . . 9 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑇 ∪ {𝐴}) = 𝑇)
2318, 22syl5eq 2811 . . . . . . . 8 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = 𝑇)
2423unieqd 4606 . . . . . . 7 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = 𝑇)
2524, 5eqtrd 2799 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (𝐹𝑈))
2617, 25syl5eqr 2813 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝑇 ∖ {𝐴}) ∪ {𝐴}) = (𝐹𝑈))
2716, 26eqtr3d 2801 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝑇 ∖ {𝐴}) ∪ 𝐴) = (𝐹𝑈))
28 difss 3901 . . . . . . 7 (𝑇 ∖ {𝐴}) ⊆ 𝑇
2928unissi 4621 . . . . . 6 (𝑇 ∖ {𝐴}) ⊆ 𝑇
3029, 5syl5sseq 3815 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑇 ∖ {𝐴}) ⊆ (𝐹𝑈))
31 uniiun 4731 . . . . . . . 8 (𝑇 ∖ {𝐴}) = 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥
3231ineq2i 3975 . . . . . . 7 (𝐴 (𝑇 ∖ {𝐴})) = (𝐴 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥)
33 incom 3969 . . . . . . 7 ( (𝑇 ∖ {𝐴}) ∩ 𝐴) = (𝐴 (𝑇 ∖ {𝐴}))
34 iunin2 4742 . . . . . . 7 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥) = (𝐴 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥)
3532, 33, 343eqtr4i 2797 . . . . . 6 ( (𝑇 ∖ {𝐴}) ∩ 𝐴) = 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥)
36 eldifsn 4474 . . . . . . . . . 10 (𝑥 ∈ (𝑇 ∖ {𝐴}) ↔ (𝑥𝑇𝑥𝐴))
37 nesym 2993 . . . . . . . . . . . 12 (𝑥𝐴 ↔ ¬ 𝐴 = 𝑥)
383cvmsdisj 31703 . . . . . . . . . . . . . 14 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝑥𝑇) → (𝐴 = 𝑥 ∨ (𝐴𝑥) = ∅))
39383expa 1147 . . . . . . . . . . . . 13 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (𝐴 = 𝑥 ∨ (𝐴𝑥) = ∅))
4039ord 890 . . . . . . . . . . . 12 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (¬ 𝐴 = 𝑥 → (𝐴𝑥) = ∅))
4137, 40syl5bi 233 . . . . . . . . . . 11 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (𝑥𝐴 → (𝐴𝑥) = ∅))
4241impr 446 . . . . . . . . . 10 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ (𝑥𝑇𝑥𝐴)) → (𝐴𝑥) = ∅)
4336, 42sylan2b 587 . . . . . . . . 9 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥 ∈ (𝑇 ∖ {𝐴})) → (𝐴𝑥) = ∅)
4443iuneq2dv 4700 . . . . . . . 8 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥) = 𝑥 ∈ (𝑇 ∖ {𝐴})∅)
45443adant1 1160 . . . . . . 7 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥) = 𝑥 ∈ (𝑇 ∖ {𝐴})∅)
46 iun0 4734 . . . . . . 7 𝑥 ∈ (𝑇 ∖ {𝐴})∅ = ∅
4745, 46syl6eq 2815 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥) = ∅)
4835, 47syl5eq 2811 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∅)
49 uneqdifeq 4219 . . . . 5 (( (𝑇 ∖ {𝐴}) ⊆ (𝐹𝑈) ∧ ( (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∅) → (( (𝑇 ∖ {𝐴}) ∪ 𝐴) = (𝐹𝑈) ↔ ((𝐹𝑈) ∖ (𝑇 ∖ {𝐴})) = 𝐴))
5030, 48, 49syl2anc 579 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (( (𝑇 ∖ {𝐴}) ∪ 𝐴) = (𝐹𝑈) ↔ ((𝐹𝑈) ∖ (𝑇 ∖ {𝐴})) = 𝐴))
5127, 50mpbid 223 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝐹𝑈) ∖ (𝑇 ∖ {𝐴})) = 𝐴)
5213, 51eqtr3d 2801 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝐶t (𝐹𝑈)) ∖ (𝑇 ∖ {𝐴})) = 𝐴)
53 uniexg 7155 . . . . . 6 (𝑇 ∈ (𝑆𝑈) → 𝑇 ∈ V)
54533ad2ant2 1164 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇 ∈ V)
555, 54eqeltrrd 2845 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝑈) ∈ V)
56 resttop 21247 . . . 4 ((𝐶 ∈ Top ∧ (𝐹𝑈) ∈ V) → (𝐶t (𝐹𝑈)) ∈ Top)
572, 55, 56syl2anc 579 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐶t (𝐹𝑈)) ∈ Top)
58 elssuni 4627 . . . . . . . . . . 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 3803 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝑥 ⊆ (𝐹𝑈))
62 df-ss 3748 . . . . . . . . 9 (𝑥 ⊆ (𝐹𝑈) ↔ (𝑥 ∩ (𝐹𝑈)) = 𝑥)
6361, 62sylib 209 . . . . . . . 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 3763 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝑥𝐶)
67 elrestr 16358 . . . . . . . . 9 ((𝐶 ∈ Top ∧ (𝐹𝑈) ∈ V ∧ 𝑥𝐶) → (𝑥 ∩ (𝐹𝑈)) ∈ (𝐶t (𝐹𝑈)))
6864, 65, 66, 67syl3anc 1490 . . . . . . . 8 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (𝑥 ∩ (𝐹𝑈)) ∈ (𝐶t (𝐹𝑈)))
6963, 68eqeltrrd 2845 . . . . . . 7 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝑥 ∈ (𝐶t (𝐹𝑈)))
7069ex 401 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑥𝑇𝑥 ∈ (𝐶t (𝐹𝑈))))
7170ssrdv 3769 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇 ⊆ (𝐶t (𝐹𝑈)))
7271ssdifssd 3912 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑇 ∖ {𝐴}) ⊆ (𝐶t (𝐹𝑈)))
73 uniopn 20984 . . . 4 (((𝐶t (𝐹𝑈)) ∈ Top ∧ (𝑇 ∖ {𝐴}) ⊆ (𝐶t (𝐹𝑈))) → (𝑇 ∖ {𝐴}) ∈ (𝐶t (𝐹𝑈)))
7457, 72, 73syl2anc 579 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑇 ∖ {𝐴}) ∈ (𝐶t (𝐹𝑈)))
75 eqid 2765 . . . 4 (𝐶t (𝐹𝑈)) = (𝐶t (𝐹𝑈))
7675opncld 21120 . . 3 (((𝐶t (𝐹𝑈)) ∈ Top ∧ (𝑇 ∖ {𝐴}) ∈ (𝐶t (𝐹𝑈))) → ( (𝐶t (𝐹𝑈)) ∖ (𝑇 ∖ {𝐴})) ∈ (Clsd‘(𝐶t (𝐹𝑈))))
7757, 74, 76syl2anc 579 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝐶t (𝐹𝑈)) ∖ (𝑇 ∖ {𝐴})) ∈ (Clsd‘(𝐶t (𝐹𝑈))))
7852, 77eqeltrrd 2845 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 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  {crab 3059  Vcvv 3350  cdif 3731  cun 3732  cin 3733  wss 3734  c0 4081  𝒫 cpw 4317  {csn 4336   cuni 4596   ciun 4678  cmpt 4890  ccnv 5278  cres 5281  cima 5282  cfv 6070  (class class class)co 6844  t crest 16350  Topctop 20980  Clsdccld 21103  Homeochmeo 21839   CovMap ccvm 31688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-oadd 7770  df-er 7949  df-en 8163  df-fin 8166  df-fi 8526  df-rest 16352  df-topgen 16373  df-top 20981  df-topon 20998  df-bases 21033  df-cld 21106  df-cvm 31689
This theorem is referenced by:  cvmliftmolem1  31714  cvmlift2lem9  31744  cvmlift3lem6  31757
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