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Theorem cvmscld 30963
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 30950 . . . . . 6 (𝐹 ∈ (𝐶 CovMap 𝐽) → 𝐶 ∈ Top)
213ad2ant1 1080 . . . . 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 30959 . . . . . . 7 (𝑇 ∈ (𝑆𝑈) → 𝑇 = (𝐹𝑈))
543ad2ant2 1081 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇 = (𝐹𝑈))
63cvmsss 30957 . . . . . . . 8 (𝑇 ∈ (𝑆𝑈) → 𝑇𝐶)
763ad2ant2 1081 . . . . . . 7 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇𝐶)
87unissd 4428 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇 𝐶)
95, 8eqsstr3d 3619 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝑈) ⊆ 𝐶)
10 eqid 2621 . . . . . 6 𝐶 = 𝐶
1110restuni 20876 . . . . 5 ((𝐶 ∈ Top ∧ (𝐹𝑈) ⊆ 𝐶) → (𝐹𝑈) = (𝐶t (𝐹𝑈)))
122, 9, 11syl2anc 692 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝑈) = (𝐶t (𝐹𝑈)))
1312difeq1d 3705 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝐹𝑈) ∖ (𝑇 ∖ {𝐴})) = ( (𝐶t (𝐹𝑈)) ∖ (𝑇 ∖ {𝐴})))
14 unisng 4418 . . . . . . 7 (𝐴𝑇 {𝐴} = 𝐴)
15143ad2ant3 1082 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → {𝐴} = 𝐴)
1615uneq2d 3745 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝑇 ∖ {𝐴}) ∪ {𝐴}) = ( (𝑇 ∖ {𝐴}) ∪ 𝐴))
17 uniun 4422 . . . . . 6 ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = ( (𝑇 ∖ {𝐴}) ∪ {𝐴})
18 undif1 4015 . . . . . . . . 9 ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (𝑇 ∪ {𝐴})
19 simp3 1061 . . . . . . . . . . 11 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝐴𝑇)
2019snssd 4309 . . . . . . . . . 10 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → {𝐴} ⊆ 𝑇)
21 ssequn2 3764 . . . . . . . . . 10 ({𝐴} ⊆ 𝑇 ↔ (𝑇 ∪ {𝐴}) = 𝑇)
2220, 21sylib 208 . . . . . . . . 9 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑇 ∪ {𝐴}) = 𝑇)
2318, 22syl5eq 2667 . . . . . . . 8 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = 𝑇)
2423unieqd 4412 . . . . . . 7 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = 𝑇)
2524, 5eqtrd 2655 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ((𝑇 ∖ {𝐴}) ∪ {𝐴}) = (𝐹𝑈))
2617, 25syl5eqr 2669 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝑇 ∖ {𝐴}) ∪ {𝐴}) = (𝐹𝑈))
2716, 26eqtr3d 2657 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝑇 ∖ {𝐴}) ∪ 𝐴) = (𝐹𝑈))
28 difss 3715 . . . . . . 7 (𝑇 ∖ {𝐴}) ⊆ 𝑇
2928unissi 4427 . . . . . 6 (𝑇 ∖ {𝐴}) ⊆ 𝑇
3029, 5syl5sseq 3632 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑇 ∖ {𝐴}) ⊆ (𝐹𝑈))
31 uniiun 4539 . . . . . . . 8 (𝑇 ∖ {𝐴}) = 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥
3231ineq2i 3789 . . . . . . 7 (𝐴 (𝑇 ∖ {𝐴})) = (𝐴 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥)
33 incom 3783 . . . . . . 7 ( (𝑇 ∖ {𝐴}) ∩ 𝐴) = (𝐴 (𝑇 ∖ {𝐴}))
34 iunin2 4550 . . . . . . 7 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥) = (𝐴 𝑥 ∈ (𝑇 ∖ {𝐴})𝑥)
3532, 33, 343eqtr4i 2653 . . . . . 6 ( (𝑇 ∖ {𝐴}) ∩ 𝐴) = 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥)
36 eldifsn 4287 . . . . . . . . . 10 (𝑥 ∈ (𝑇 ∖ {𝐴}) ↔ (𝑥𝑇𝑥𝐴))
37 nesym 2846 . . . . . . . . . . . 12 (𝑥𝐴 ↔ ¬ 𝐴 = 𝑥)
383cvmsdisj 30960 . . . . . . . . . . . . . 14 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇𝑥𝑇) → (𝐴 = 𝑥 ∨ (𝐴𝑥) = ∅))
39383expa 1262 . . . . . . . . . . . . 13 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (𝐴 = 𝑥 ∨ (𝐴𝑥) = ∅))
4039ord 392 . . . . . . . . . . . 12 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (¬ 𝐴 = 𝑥 → (𝐴𝑥) = ∅))
4137, 40syl5bi 232 . . . . . . . . . . 11 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (𝑥𝐴 → (𝐴𝑥) = ∅))
4241impr 648 . . . . . . . . . 10 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ (𝑥𝑇𝑥𝐴)) → (𝐴𝑥) = ∅)
4336, 42sylan2b 492 . . . . . . . . 9 (((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥 ∈ (𝑇 ∖ {𝐴})) → (𝐴𝑥) = ∅)
4443iuneq2dv 4508 . . . . . . . 8 ((𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥) = 𝑥 ∈ (𝑇 ∖ {𝐴})∅)
45443adant1 1077 . . . . . . 7 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥) = 𝑥 ∈ (𝑇 ∖ {𝐴})∅)
46 iun0 4542 . . . . . . 7 𝑥 ∈ (𝑇 ∖ {𝐴})∅ = ∅
4745, 46syl6eq 2671 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑥 ∈ (𝑇 ∖ {𝐴})(𝐴𝑥) = ∅)
4835, 47syl5eq 2667 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∅)
49 uneqdifeq 4029 . . . . 5 (( (𝑇 ∖ {𝐴}) ⊆ (𝐹𝑈) ∧ ( (𝑇 ∖ {𝐴}) ∩ 𝐴) = ∅) → (( (𝑇 ∖ {𝐴}) ∪ 𝐴) = (𝐹𝑈) ↔ ((𝐹𝑈) ∖ (𝑇 ∖ {𝐴})) = 𝐴))
5030, 48, 49syl2anc 692 . . . 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 2657 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝐶t (𝐹𝑈)) ∖ (𝑇 ∖ {𝐴})) = 𝐴)
53 uniexg 6908 . . . . . 6 (𝑇 ∈ (𝑆𝑈) → 𝑇 ∈ V)
54533ad2ant2 1081 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇 ∈ V)
555, 54eqeltrrd 2699 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐹𝑈) ∈ V)
56 resttop 20874 . . . 4 ((𝐶 ∈ Top ∧ (𝐹𝑈) ∈ V) → (𝐶t (𝐹𝑈)) ∈ Top)
572, 55, 56syl2anc 692 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝐶t (𝐹𝑈)) ∈ Top)
58 elssuni 4433 . . . . . . . . . . 11 (𝑥𝑇𝑥 𝑇)
5958adantl 482 . . . . . . . . . 10 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝑥 𝑇)
605adantr 481 . . . . . . . . . 10 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝑇 = (𝐹𝑈))
6159, 60sseqtrd 3620 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝑥 ⊆ (𝐹𝑈))
62 df-ss 3569 . . . . . . . . 9 (𝑥 ⊆ (𝐹𝑈) ↔ (𝑥 ∩ (𝐹𝑈)) = 𝑥)
6361, 62sylib 208 . . . . . . . 8 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (𝑥 ∩ (𝐹𝑈)) = 𝑥)
642adantr 481 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝐶 ∈ Top)
6555adantr 481 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (𝐹𝑈) ∈ V)
667sselda 3583 . . . . . . . . 9 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝑥𝐶)
67 elrestr 16010 . . . . . . . . 9 ((𝐶 ∈ Top ∧ (𝐹𝑈) ∈ V ∧ 𝑥𝐶) → (𝑥 ∩ (𝐹𝑈)) ∈ (𝐶t (𝐹𝑈)))
6864, 65, 66, 67syl3anc 1323 . . . . . . . 8 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → (𝑥 ∩ (𝐹𝑈)) ∈ (𝐶t (𝐹𝑈)))
6963, 68eqeltrrd 2699 . . . . . . 7 (((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) ∧ 𝑥𝑇) → 𝑥 ∈ (𝐶t (𝐹𝑈)))
7069ex 450 . . . . . 6 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑥𝑇𝑥 ∈ (𝐶t (𝐹𝑈))))
7170ssrdv 3589 . . . . 5 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → 𝑇 ⊆ (𝐶t (𝐹𝑈)))
7271ssdifssd 3726 . . . 4 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑇 ∖ {𝐴}) ⊆ (𝐶t (𝐹𝑈)))
73 uniopn 20627 . . . 4 (((𝐶t (𝐹𝑈)) ∈ Top ∧ (𝑇 ∖ {𝐴}) ⊆ (𝐶t (𝐹𝑈))) → (𝑇 ∖ {𝐴}) ∈ (𝐶t (𝐹𝑈)))
7457, 72, 73syl2anc 692 . . 3 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → (𝑇 ∖ {𝐴}) ∈ (𝐶t (𝐹𝑈)))
75 eqid 2621 . . . 4 (𝐶t (𝐹𝑈)) = (𝐶t (𝐹𝑈))
7675opncld 20747 . . 3 (((𝐶t (𝐹𝑈)) ∈ Top ∧ (𝑇 ∖ {𝐴}) ∈ (𝐶t (𝐹𝑈))) → ( (𝐶t (𝐹𝑈)) ∖ (𝑇 ∖ {𝐴})) ∈ (Clsd‘(𝐶t (𝐹𝑈))))
7757, 74, 76syl2anc 692 . 2 ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑇 ∈ (𝑆𝑈) ∧ 𝐴𝑇) → ( (𝐶t (𝐹𝑈)) ∖ (𝑇 ∖ {𝐴})) ∈ (Clsd‘(𝐶t (𝐹𝑈))))
7852, 77eqeltrrd 2699 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 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  {crab 2911  Vcvv 3186  cdif 3552  cun 3553  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148   cuni 4402   ciun 4485  cmpt 4673  ccnv 5073  cres 5076  cima 5077  cfv 5847  (class class class)co 6604  t crest 16002  Topctop 20617  Clsdccld 20730  Homeochmeo 21466   CovMap ccvm 30945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-er 7687  df-en 7900  df-fin 7903  df-fi 8261  df-rest 16004  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-cld 20733  df-cvm 30946
This theorem is referenced by:  cvmliftmolem1  30971  cvmlift2lem9  31001  cvmlift3lem6  31014
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