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Theorem cvmsval 31580
Description: Elementhood in the set 𝑆 of all even coverings of an open set in 𝐽. 𝑆 is an even covering of 𝑈 if it is a nonempty collection of disjoint open sets in 𝐶 whose union is the preimage of 𝑈, such that each set 𝑢𝑆 is homeomorphic under 𝐹 to 𝑈. (Contributed by Mario Carneiro, 13-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
cvmsval (𝐶𝑉 → (𝑇 ∈ (𝑆𝑈) ↔ (𝑈𝐽 ∧ (𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 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   𝑘,𝑉,𝑠,𝑢,𝑣
Allowed substitution hints:   𝑆(𝑣,𝑢,𝑘,𝑠)   𝑇(𝑘)

Proof of Theorem cvmsval
StepHypRef Expression
1 cvmcov.1 . . 3 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
21cvmsi 31579 . 2 (𝑇 ∈ (𝑆𝑈) → (𝑈𝐽 ∧ (𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))))
3 3anass 1079 . . 3 ((𝑈𝐽 ∧ (𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))) ↔ (𝑈𝐽 ∧ ((𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))))
4 id 22 . . . . . . . 8 (𝑈𝐽𝑈𝐽)
5 pwexg 4978 . . . . . . . . 9 (𝐶𝑉 → 𝒫 𝐶 ∈ V)
6 difexg 4939 . . . . . . . . 9 (𝒫 𝐶 ∈ V → (𝒫 𝐶 ∖ {∅}) ∈ V)
7 rabexg 4942 . . . . . . . . 9 ((𝒫 𝐶 ∖ {∅}) ∈ V → {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))} ∈ V)
85, 6, 73syl 18 . . . . . . . 8 (𝐶𝑉 → {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))} ∈ V)
9 imaeq2 5603 . . . . . . . . . . . 12 (𝑘 = 𝑈 → (𝐹𝑘) = (𝐹𝑈))
109eqeq2d 2780 . . . . . . . . . . 11 (𝑘 = 𝑈 → ( 𝑠 = (𝐹𝑘) ↔ 𝑠 = (𝐹𝑈)))
11 oveq2 6800 . . . . . . . . . . . . . . 15 (𝑘 = 𝑈 → (𝐽t 𝑘) = (𝐽t 𝑈))
1211oveq2d 6808 . . . . . . . . . . . . . 14 (𝑘 = 𝑈 → ((𝐶t 𝑢)Homeo(𝐽t 𝑘)) = ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))
1312eleq2d 2835 . . . . . . . . . . . . 13 (𝑘 = 𝑈 → ((𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)) ↔ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))
1413anbi2d 606 . . . . . . . . . . . 12 (𝑘 = 𝑈 → ((∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))) ↔ (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))
1514ralbidv 3134 . . . . . . . . . . 11 (𝑘 = 𝑈 → (∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))) ↔ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))
1610, 15anbi12d 608 . . . . . . . . . 10 (𝑘 = 𝑈 → (( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))) ↔ ( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))))
1716rabbidv 3338 . . . . . . . . 9 (𝑘 = 𝑈 → {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))} = {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))})
1817, 1fvmptg 6422 . . . . . . . 8 ((𝑈𝐽 ∧ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))} ∈ V) → (𝑆𝑈) = {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))})
194, 8, 18syl2anr 576 . . . . . . 7 ((𝐶𝑉𝑈𝐽) → (𝑆𝑈) = {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))})
2019eleq2d 2835 . . . . . 6 ((𝐶𝑉𝑈𝐽) → (𝑇 ∈ (𝑆𝑈) ↔ 𝑇 ∈ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))}))
21 unieq 4580 . . . . . . . . . 10 (𝑠 = 𝑇 𝑠 = 𝑇)
2221eqeq1d 2772 . . . . . . . . 9 (𝑠 = 𝑇 → ( 𝑠 = (𝐹𝑈) ↔ 𝑇 = (𝐹𝑈)))
23 difeq1 3870 . . . . . . . . . . . 12 (𝑠 = 𝑇 → (𝑠 ∖ {𝑢}) = (𝑇 ∖ {𝑢}))
2423raleqdv 3292 . . . . . . . . . . 11 (𝑠 = 𝑇 → (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ↔ ∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅))
2524anbi1d 607 . . . . . . . . . 10 (𝑠 = 𝑇 → ((∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))) ↔ (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))
2625raleqbi1dv 3294 . . . . . . . . 9 (𝑠 = 𝑇 → (∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))) ↔ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))
2722, 26anbi12d 608 . . . . . . . 8 (𝑠 = 𝑇 → (( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))) ↔ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))))
2827elrab 3513 . . . . . . 7 (𝑇 ∈ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))} ↔ (𝑇 ∈ (𝒫 𝐶 ∖ {∅}) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))))
29 eldifsn 4451 . . . . . . . . 9 (𝑇 ∈ (𝒫 𝐶 ∖ {∅}) ↔ (𝑇 ∈ 𝒫 𝐶𝑇 ≠ ∅))
30 elpw2g 4955 . . . . . . . . . . 11 (𝐶𝑉 → (𝑇 ∈ 𝒫 𝐶𝑇𝐶))
3130adantr 466 . . . . . . . . . 10 ((𝐶𝑉𝑈𝐽) → (𝑇 ∈ 𝒫 𝐶𝑇𝐶))
3231anbi1d 607 . . . . . . . . 9 ((𝐶𝑉𝑈𝐽) → ((𝑇 ∈ 𝒫 𝐶𝑇 ≠ ∅) ↔ (𝑇𝐶𝑇 ≠ ∅)))
3329, 32syl5bb 272 . . . . . . . 8 ((𝐶𝑉𝑈𝐽) → (𝑇 ∈ (𝒫 𝐶 ∖ {∅}) ↔ (𝑇𝐶𝑇 ≠ ∅)))
3433anbi1d 607 . . . . . . 7 ((𝐶𝑉𝑈𝐽) → ((𝑇 ∈ (𝒫 𝐶 ∖ {∅}) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))) ↔ ((𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))))
3528, 34syl5bb 272 . . . . . 6 ((𝐶𝑉𝑈𝐽) → (𝑇 ∈ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))} ↔ ((𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))))
3620, 35bitrd 268 . . . . 5 ((𝐶𝑉𝑈𝐽) → (𝑇 ∈ (𝑆𝑈) ↔ ((𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))))
3736biimprd 238 . . . 4 ((𝐶𝑉𝑈𝐽) → (((𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))) → 𝑇 ∈ (𝑆𝑈)))
3837expimpd 441 . . 3 (𝐶𝑉 → ((𝑈𝐽 ∧ ((𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))) → 𝑇 ∈ (𝑆𝑈)))
393, 38syl5bi 232 . 2 (𝐶𝑉 → ((𝑈𝐽 ∧ (𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))) → 𝑇 ∈ (𝑆𝑈)))
402, 39impbid2 216 1 (𝐶𝑉 → (𝑇 ∈ (𝑆𝑈) ↔ (𝑈𝐽 ∧ (𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wne 2942  wral 3060  {crab 3064  Vcvv 3349  cdif 3718  cin 3720  wss 3721  c0 4061  𝒫 cpw 4295  {csn 4314   cuni 4572  cmpt 4861  ccnv 5248  cres 5251  cima 5252  cfv 6031  (class class class)co 6792  t crest 16288  Homeochmeo 21776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795
This theorem is referenced by:  cvmsss2  31588
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