Theorem cvmsval 36031

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 𝑢 ∈ 𝑆 is homeomorphic under 𝐹 to 𝑈. (Contributed by Mario Carneiro, 13-Feb-2015.)

Hypothesis

Ref	Expression
cvmcov.1	⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})

Assertion

Ref	Expression
cvmsval	⊢ (𝐶 ∈ 𝑉 → (𝑇 ∈ (𝑆‘𝑈) ↔ (𝑈 ∈ 𝐽 ∧ (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 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

Step	Hyp	Ref	Expression
1		cvmcov.1	. . 3 ⊢ 𝑆 = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})
2	1	cvmsi 36030	. 2 ⊢ (𝑇 ∈ (𝑆‘𝑈) → (𝑈 ∈ 𝐽 ∧ (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))))
3		3anass 1111	. . 3 ⊢ ((𝑈 ∈ 𝐽 ∧ (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))) ↔ (𝑈 ∈ 𝐽 ∧ ((𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))))
4		id 23	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐽 → 𝑈 ∈ 𝐽)
5		pwexg 5339	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝑉 → 𝒫 𝐶 ∈ V)
6		difexg 5290	. . . . . . . . 9 ⊢ (𝒫 𝐶 ∈ V → (𝒫 𝐶 ∖ {∅}) ∈ V)
7		rabexg 5298	. . . . . . . . 9 ⊢ ((𝒫 𝐶 ∖ {∅}) ∈ V → {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))} ∈ V)
8	5, 6, 7	3syl 19	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑉 → {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))} ∈ V)
9		imaeq2 6047	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑈 → (◡𝐹 “ 𝑘) = (◡𝐹 “ 𝑈))
10	9	eqeq2d 2771	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑈 → (∪ 𝑠 = (◡𝐹 “ 𝑘) ↔ ∪ 𝑠 = (◡𝐹 “ 𝑈)))
11		oveq2 7428	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑈 → (𝐽 ↾t 𝑘) = (𝐽 ↾t 𝑈))
12	11	oveq2d 7436	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑈 → ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)) = ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))
13	12	eleq2d 2846	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑈 → ((𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)) ↔ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))
14	13	anbi2d 642	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑈 → ((∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))) ↔ (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))
15	14	ralbidv 3185	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑈 → (∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))) ↔ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))
16	10, 15	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑘 = 𝑈 → ((∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))) ↔ (∪ 𝑠 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))))
17	16	rabbidv 3419	. . . . . . . . 9 ⊢ (𝑘 = 𝑈 → {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))} = {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))})
18	17, 1	fvmptg 6991	. . . . . . . 8 ⊢ ((𝑈 ∈ 𝐽 ∧ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))} ∈ V) → (𝑆‘𝑈) = {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))})
19	4, 8, 18	syl2anr 609	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑈 ∈ 𝐽) → (𝑆‘𝑈) = {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))})
20	19	eleq2d 2846	. . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑈 ∈ 𝐽) → (𝑇 ∈ (𝑆‘𝑈) ↔ 𝑇 ∈ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))}))
21		unieq 4877	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → ∪ 𝑠 = ∪ 𝑇)
22	21	eqeq1d 2762	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → (∪ 𝑠 = (◡𝐹 “ 𝑈) ↔ ∪ 𝑇 = (◡𝐹 “ 𝑈)))
23		difeq1 4056	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑇 → (𝑠 ∖ {𝑢}) = (𝑇 ∖ {𝑢}))
24	23	raleqdv 3319	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑇 → (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ↔ ∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅))
25	24	anbi1d 643	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → ((∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))) ↔ (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))
26	25	raleqbi1dv 3329	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → (∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))) ↔ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))
27	22, 26	anbi12d 644	. . . . . . . 8 ⊢ (𝑠 = 𝑇 → ((∪ 𝑠 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))) ↔ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))))
28	27	elrab 3644	. . . . . . 7 ⊢ (𝑇 ∈ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))} ↔ (𝑇 ∈ (𝒫 𝐶 ∖ {∅}) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))))
29		eldifsn 4747	. . . . . . . . 9 ⊢ (𝑇 ∈ (𝒫 𝐶 ∖ {∅}) ↔ (𝑇 ∈ 𝒫 𝐶 ∧ 𝑇 ≠ ∅))
30		elpw2g 5294	. . . . . . . . . . 11 ⊢ (𝐶 ∈ 𝑉 → (𝑇 ∈ 𝒫 𝐶 ↔ 𝑇 ⊆ 𝐶))
31	30	adantr 486	. . . . . . . . . 10 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑈 ∈ 𝐽) → (𝑇 ∈ 𝒫 𝐶 ↔ 𝑇 ⊆ 𝐶))
32	31	anbi1d 643	. . . . . . . . 9 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑈 ∈ 𝐽) → ((𝑇 ∈ 𝒫 𝐶 ∧ 𝑇 ≠ ∅) ↔ (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅)))
33	29, 32	bitrid 286	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑈 ∈ 𝐽) → (𝑇 ∈ (𝒫 𝐶 ∖ {∅}) ↔ (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅)))
34	33	anbi1d 643	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑈 ∈ 𝐽) → ((𝑇 ∈ (𝒫 𝐶 ∖ {∅}) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))) ↔ ((𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))))
35	28, 34	bitrid 286	. . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑈 ∈ 𝐽) → (𝑇 ∈ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))} ↔ ((𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))))
36	20, 35	bitrd 282	. . . . 5 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑈 ∈ 𝐽) → (𝑇 ∈ (𝑆‘𝑈) ↔ ((𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))))
37	36	biimprd 251	. . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝑈 ∈ 𝐽) → (((𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))) → 𝑇 ∈ (𝑆‘𝑈)))
38	37	expimpd 459	. . 3 ⊢ (𝐶 ∈ 𝑉 → ((𝑈 ∈ 𝐽 ∧ ((𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))) → 𝑇 ∈ (𝑆‘𝑈)))
39	3, 38	biimtrid 245	. 2 ⊢ (𝐶 ∈ 𝑉 → ((𝑈 ∈ 𝐽 ∧ (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈))))) → 𝑇 ∈ (𝑆‘𝑈)))
40	2, 39	impbid2 229	1 ⊢ (𝐶 ∈ 𝑉 → (𝑇 ∈ (𝑆‘𝑈) ↔ (𝑈 ∈ 𝐽 ∧ (𝑇 ⊆ 𝐶 ∧ 𝑇 ≠ ∅) ∧ (∪ 𝑇 = (◡𝐹 “ 𝑈) ∧ ∀𝑢 ∈ 𝑇 (∀𝑣 ∈ (𝑇 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑈)))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089  ∅c0 4278  𝒫 cpw 4556  {csn 4583  ∪ cuni 4866   ↦ cmpt 5185  ^◡ccnv 5649   ↾ cres 5652   “ cima 5653  ‘cfv 6538  (class class class)co 7420   ↾_t crest 17591  Homeochmeo 24072

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423

This theorem is used by:  cvmsss2  36039