Theorem iscvm 35324

Description: The property of being a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)

Hypotheses

Ref	Expression
iscvm.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))))})
iscvm.2	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
iscvm	⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)))

Distinct variable groups:   𝑘,𝑠,𝑢,𝑣,𝑥   𝐶,𝑘,𝑠,𝑢,𝑥   𝑥,𝑋   𝑘,𝐹,𝑠,𝑢,𝑥   𝑘,𝐽,𝑠,𝑢,𝑥

Allowed substitution hints:   𝐶(𝑣)   𝑆(𝑥,𝑣,𝑢,𝑘,𝑠)   𝐹(𝑣)   𝐽(𝑣)   𝑋(𝑣,𝑢,𝑘,𝑠)

Proof of Theorem iscvm

Dummy variables 𝑐 𝑓 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		anass 468	. 2 ⊢ ((((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ (𝐹 ∈ (𝐶 Cn 𝐽) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅))))
2		df-3an 1088	. . 3 ⊢ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ 𝐹 ∈ (𝐶 Cn 𝐽)))
3	2	anbi1i 624	. 2 ⊢ (((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)) ↔ (((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)))
4		df-cvm 35321	. . . 4 ⊢ CovMap = (𝑐 ∈ Top, 𝑗 ∈ Top ↦ {𝑓 ∈ (𝑐 Cn 𝑗) ∣ ∀𝑥 ∈ ∪ 𝑗∃𝑘 ∈ 𝑗 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝑐 ↾t 𝑢)Homeo(𝑗 ↾t 𝑘)))))})
5	4	elmpocl 7593	. . 3 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝐶 ∈ Top ∧ 𝐽 ∈ Top))
6		oveq12 7361	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → (𝑐 Cn 𝑗) = (𝐶 Cn 𝐽))
7		simpr 484	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
8	7	unieqd 4871	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = ∪ 𝐽)
9		iscvm.2	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽
10	8, 9	eqtr4di 2786	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋)
11		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → 𝑐 = 𝐶)
12	11	pweqd 4566	. . . . . . . . . . . 12 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → 𝒫 𝑐 = 𝒫 𝐶)
13	12	difeq1d 4074	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → (𝒫 𝑐 ∖ {∅}) = (𝒫 𝐶 ∖ {∅}))
14		oveq1 7359	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝐶 → (𝑐 ↾t 𝑢) = (𝐶 ↾t 𝑢))
15		oveq1 7359	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → (𝑗 ↾t 𝑘) = (𝐽 ↾t 𝑘))
16	14, 15	oveqan12d 7371	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → ((𝑐 ↾t 𝑢)Homeo(𝑗 ↾t 𝑘)) = ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))
17	16	eleq2d 2819	. . . . . . . . . . . . . 14 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → ((𝑓 ↾ 𝑢) ∈ ((𝑐 ↾t 𝑢)Homeo(𝑗 ↾t 𝑘)) ↔ (𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))
18	17	anbi2d 630	. . . . . . . . . . . . 13 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → ((∀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 𝑘)))))
19	18	ralbidv 3156	. . . . . . . . . . . 12 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → (∀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 𝑘)))))
20	19	anbi2d 630	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → ((∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝑐 ↾t 𝑢)Homeo(𝑗 ↾t 𝑘)))) ↔ (∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))))
21	13, 20	rexeqbidv 3314	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → (∃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 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))))
22	21	anbi2d 630	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → ((𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝑐 ↾t 𝑢)Homeo(𝑗 ↾t 𝑘))))) ↔ (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))))))
23	7, 22	rexeqbidv 3314	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → (∃𝑘 ∈ 𝑗 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝑐 ↾t 𝑢)Homeo(𝑗 ↾t 𝑘))))) ↔ ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))))))
24	10, 23	raleqbidv 3313	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → (∀𝑥 ∈ ∪ 𝑗∃𝑘 ∈ 𝑗 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝑐 ↾t 𝑢)Homeo(𝑗 ↾t 𝑘))))) ↔ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))))))
25	6, 24	rabeqbidv 3414	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → {𝑓 ∈ (𝑐 Cn 𝑗) ∣ ∀𝑥 ∈ ∪ 𝑗∃𝑘 ∈ 𝑗 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝑐 ↾t 𝑢)Homeo(𝑗 ↾t 𝑘)))))} = {𝑓 ∈ (𝐶 Cn 𝐽) ∣ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))))})
26		ovex 7385	. . . . . . 7 ⊢ (𝐶 Cn 𝐽) ∈ V
27	26	rabex 5279	. . . . . 6 ⊢ {𝑓 ∈ (𝐶 Cn 𝐽) ∣ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))))} ∈ V
28	25, 4, 27	ovmpoa 7507	. . . . 5 ⊢ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (𝐶 CovMap 𝐽) = {𝑓 ∈ (𝐶 Cn 𝐽) ∣ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))))})
29	28	eleq2d 2819	. . . 4 ⊢ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ 𝐹 ∈ {𝑓 ∈ (𝐶 Cn 𝐽) ∣ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))))}))
30		id 22	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝐽 → 𝑘 ∈ 𝐽)
31		pwexg 5318	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ Top → 𝒫 𝐶 ∈ V)
32	31	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → 𝒫 𝐶 ∈ V)
33		difexg 5269	. . . . . . . . . . . . 13 ⊢ (𝒫 𝐶 ∈ V → (𝒫 𝐶 ∖ {∅}) ∈ V)
34		rabexg 5277	. . . . . . . . . . . . 13 ⊢ ((𝒫 𝐶 ∖ {∅}) ∈ V → {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))} ∈ V)
35	32, 33, 34	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))} ∈ V)
36		iscvm.1	. . . . . . . . . . . . 13 ⊢ 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))))})
37	36	fvmpt2 6946	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ 𝐽 ∧ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))} ∈ V) → (𝑆‘𝑘) = {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})
38	30, 35, 37	syl2anr 597	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ 𝑘 ∈ 𝐽) → (𝑆‘𝑘) = {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))})
39	38	neeq1d 2988	. . . . . . . . . 10 ⊢ (((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ 𝑘 ∈ 𝐽) → ((𝑆‘𝑘) ≠ ∅ ↔ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))} ≠ ∅))
40		rabn0 4338	. . . . . . . . . 10 ⊢ ({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 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))))
41	39, 40	bitrdi 287	. . . . . . . . 9 ⊢ (((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ 𝑘 ∈ 𝐽) → ((𝑆‘𝑘) ≠ ∅ ↔ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))))
42	41	anbi2d 630	. . . . . . . 8 ⊢ (((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ 𝑘 ∈ 𝐽) → ((𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅) ↔ (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))))))
43	42	rexbidva 3155	. . . . . . 7 ⊢ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅) ↔ ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))))))
44	43	ralbidv 3156	. . . . . 6 ⊢ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅) ↔ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))))))
45	44	anbi2d 630	. . . . 5 ⊢ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → ((𝐹 ∈ (𝐶 Cn 𝐽) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)) ↔ (𝐹 ∈ (𝐶 Cn 𝐽) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))))))
46		cnveq 5817	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
47	46	imaeq1d 6012	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ 𝑘) = (◡𝐹 “ 𝑘))
48	47	eqeq2d 2744	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (∪ 𝑠 = (◡𝑓 “ 𝑘) ↔ ∪ 𝑠 = (◡𝐹 “ 𝑘)))
49		reseq1 5926	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐹 → (𝑓 ↾ 𝑢) = (𝐹 ↾ 𝑢))
50	49	eleq1d 2818	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐹 → ((𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)) ↔ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))
51	50	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → ((∀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 𝑘)))))
52	51	ralbidv 3156	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (∀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 𝑘)))))
53	48, 52	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ((∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))) ↔ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))))
54	53	rexbidv 3157	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (∃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 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))))
55	54	anbi2d 630	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))) ↔ (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))))))
56	55	rexbidv 3157	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))) ↔ ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))))))
57	56	ralbidv 3156	. . . . . 6 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))) ↔ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))))))
58	57	elrab 3643	. . . . 5 ⊢ (𝐹 ∈ {𝑓 ∈ (𝐶 Cn 𝐽) ∣ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))))} ↔ (𝐹 ∈ (𝐶 Cn 𝐽) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))))))
59	45, 58	bitr4di 289	. . . 4 ⊢ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → ((𝐹 ∈ (𝐶 Cn 𝐽) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)) ↔ 𝐹 ∈ {𝑓 ∈ (𝐶 Cn 𝐽) ∣ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})(∪ 𝑠 = (◡𝑓 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))))}))
60	29, 59	bitr4d 282	. . 3 ⊢ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ (𝐹 ∈ (𝐶 Cn 𝐽) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅))))
61	5, 60	biadanii 821	. 2 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ (𝐹 ∈ (𝐶 Cn 𝐽) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅))))
62	1, 3, 61	3bitr4ri 304	1 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  ∃wrex 3057  {crab 3396  Vcvv 3437   ∖ cdif 3895   ∩ cin 3897  ∅c0 4282  𝒫 cpw 4549  {csn 4575  ∪ cuni 4858   ↦ cmpt 5174  ^◡ccnv 5618   ↾ cres 5621   “ cima 5622  ‘cfv 6486  (class class class)co 7352   ↾_t crest 17326  Topctop 22809   Cn ccn 23140  Homeochmeo 23669   CovMap ccvm 35320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-cvm 35321

This theorem is referenced by:  cvmcn  35327  cvmcov  35328