Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmsval Structured version   Visualization version   GIF version

Theorem cvmsval 34909
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 34908 . 2 (𝑇 ∈ (𝑆𝑈) → (𝑈𝐽 ∧ (𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))))
3 3anass 1092 . . 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 5382 . . . . . . . . 9 (𝐶𝑉 → 𝒫 𝐶 ∈ V)
6 difexg 5333 . . . . . . . . 9 (𝒫 𝐶 ∈ V → (𝒫 𝐶 ∖ {∅}) ∈ V)
7 rabexg 5337 . . . . . . . . 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 6064 . . . . . . . . . . . 12 (𝑘 = 𝑈 → (𝐹𝑘) = (𝐹𝑈))
109eqeq2d 2739 . . . . . . . . . . 11 (𝑘 = 𝑈 → ( 𝑠 = (𝐹𝑘) ↔ 𝑠 = (𝐹𝑈)))
11 oveq2 7434 . . . . . . . . . . . . . . 15 (𝑘 = 𝑈 → (𝐽t 𝑘) = (𝐽t 𝑈))
1211oveq2d 7442 . . . . . . . . . . . . . 14 (𝑘 = 𝑈 → ((𝐶t 𝑢)Homeo(𝐽t 𝑘)) = ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))
1312eleq2d 2815 . . . . . . . . . . . . 13 (𝑘 = 𝑈 → ((𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)) ↔ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))
1413anbi2d 628 . . . . . . . . . . . 12 (𝑘 = 𝑈 → ((∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))) ↔ (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))
1514ralbidv 3175 . . . . . . . . . . 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 630 . . . . . . . . . 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 3438 . . . . . . . . 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 7008 . . . . . . . 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 595 . . . . . . 7 ((𝐶𝑉𝑈𝐽) → (𝑆𝑈) = {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))})
2019eleq2d 2815 . . . . . 6 ((𝐶𝑉𝑈𝐽) → (𝑇 ∈ (𝑆𝑈) ↔ 𝑇 ∈ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑈) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))}))
21 unieq 4923 . . . . . . . . . 10 (𝑠 = 𝑇 𝑠 = 𝑇)
2221eqeq1d 2730 . . . . . . . . 9 (𝑠 = 𝑇 → ( 𝑠 = (𝐹𝑈) ↔ 𝑇 = (𝐹𝑈)))
23 difeq1 4115 . . . . . . . . . . . 12 (𝑠 = 𝑇 → (𝑠 ∖ {𝑢}) = (𝑇 ∖ {𝑢}))
2423raleqdv 3323 . . . . . . . . . . 11 (𝑠 = 𝑇 → (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ↔ ∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅))
2524anbi1d 629 . . . . . . . . . 10 (𝑠 = 𝑇 → ((∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))) ↔ (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))
2625raleqbi1dv 3331 . . . . . . . . 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 630 . . . . . . . 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 3684 . . . . . . 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 4795 . . . . . . . . 9 (𝑇 ∈ (𝒫 𝐶 ∖ {∅}) ↔ (𝑇 ∈ 𝒫 𝐶𝑇 ≠ ∅))
30 elpw2g 5350 . . . . . . . . . . 11 (𝐶𝑉 → (𝑇 ∈ 𝒫 𝐶𝑇𝐶))
3130adantr 479 . . . . . . . . . 10 ((𝐶𝑉𝑈𝐽) → (𝑇 ∈ 𝒫 𝐶𝑇𝐶))
3231anbi1d 629 . . . . . . . . 9 ((𝐶𝑉𝑈𝐽) → ((𝑇 ∈ 𝒫 𝐶𝑇 ≠ ∅) ↔ (𝑇𝐶𝑇 ≠ ∅)))
3329, 32bitrid 282 . . . . . . . 8 ((𝐶𝑉𝑈𝐽) → (𝑇 ∈ (𝒫 𝐶 ∖ {∅}) ↔ (𝑇𝐶𝑇 ≠ ∅)))
3433anbi1d 629 . . . . . . 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, 34bitrid 282 . . . . . 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 278 . . . . 5 ((𝐶𝑉𝑈𝐽) → (𝑇 ∈ (𝑆𝑈) ↔ ((𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))))
3736biimprd 247 . . . 4 ((𝐶𝑉𝑈𝐽) → (((𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))) → 𝑇 ∈ (𝑆𝑈)))
3837expimpd 452 . . 3 (𝐶𝑉 → ((𝑈𝐽 ∧ ((𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈)))))) → 𝑇 ∈ (𝑆𝑈)))
393, 38biimtrid 241 . 2 (𝐶𝑉 → ((𝑈𝐽 ∧ (𝑇𝐶𝑇 ≠ ∅) ∧ ( 𝑇 = (𝐹𝑈) ∧ ∀𝑢𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑈))))) → 𝑇 ∈ (𝑆𝑈)))
402, 39impbid2 225 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 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2937  wral 3058  {crab 3430  Vcvv 3473  cdif 3946  cin 3948  wss 3949  c0 4326  𝒫 cpw 4606  {csn 4632   cuni 4912  cmpt 5235  ccnv 5681  cres 5684  cima 5685  cfv 6553  (class class class)co 7426  t crest 17409  Homeochmeo 23677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429
This theorem is referenced by:  cvmsss2  34917
  Copyright terms: Public domain W3C validator