Theorem iscvm 31241

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 681	. 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 1039	. . 3 ⊢ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ 𝐹 ∈ (𝐶 Cn 𝐽)))
3	2	anbi1i 731	. 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 31238	. . . 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	elmpt2cl 6876	. . 3 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝐶 ∈ Top ∧ 𝐽 ∈ Top))
6		oveq12 6659	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → (𝑐 Cn 𝑗) = (𝐶 Cn 𝐽))
7		simpr 477	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽)
8	7	unieqd 4446	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = ∪ 𝐽)
9		iscvm.2	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽
10	8, 9	syl6eqr 2674	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋)
11		simpl 473	. . . . . . . . . . . . 13 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → 𝑐 = 𝐶)
12	11	pweqd 4163	. . . . . . . . . . . 12 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → 𝒫 𝑐 = 𝒫 𝐶)
13	12	difeq1d 3727	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → (𝒫 𝑐 ∖ {∅}) = (𝒫 𝐶 ∖ {∅}))
14		oveq1 6657	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝐶 → (𝑐 ↾t 𝑢) = (𝐶 ↾t 𝑢))
15		oveq1 6657	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → (𝑗 ↾t 𝑘) = (𝐽 ↾t 𝑘))
16	14, 15	oveqan12d 6669	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → ((𝑐 ↾t 𝑢)Homeo(𝑗 ↾t 𝑘)) = ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))
17	16	eleq2d 2687	. . . . . . . . . . . . . 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 740	. . . . . . . . . . . . 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 2986	. . . . . . . . . . . 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 740	. . . . . . . . . . 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 3153	. . . . . . . . . 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 740	. . . . . . . . 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 3153	. . . . . . . 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 3152	. . . . . . 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 3195	. . . . . 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 6678	. . . . . . 7 ⊢ (𝐶 Cn 𝐽) ∈ V
27	26	rabex 4813	. . . . . 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	ovmpt2a 6791	. . . . 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 2687	. . . 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 4850	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ Top → 𝒫 𝐶 ∈ V)
32	31	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → 𝒫 𝐶 ∈ V)
33		difexg 4808	. . . . . . . . . . . . 13 ⊢ (𝒫 𝐶 ∈ V → (𝒫 𝐶 ∖ {∅}) ∈ V)
34		rabexg 4812	. . . . . . . . . . . . 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 6291	. . . . . . . . . . . 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 495	. . . . . . . . . . 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 2853	. . . . . . . . . 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 3958	. . . . . . . . . 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	syl6bb 276	. . . . . . . . 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 740	. . . . . . . 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 3049	. . . . . . 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 2986	. . . . . 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 740	. . . . 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 5296	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹)
47	46	imaeq1d 5465	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ 𝑘) = (◡𝐹 “ 𝑘))
48	47	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (∪ 𝑠 = (◡𝑓 “ 𝑘) ↔ ∪ 𝑠 = (◡𝐹 “ 𝑘)))
49		reseq1 5390	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐹 → (𝑓 ↾ 𝑢) = (𝐹 ↾ 𝑢))
50	49	eleq1d 2686	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐹 → ((𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)) ↔ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))
51	50	anbi2d 740	. . . . . . . . . . . 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 2986	. . . . . . . . . . 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 747	. . . . . . . . . 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 3052	. . . . . . . . 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 740	. . . . . . . 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 3052	. . . . . . 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 2986	. . . . . 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 3363	. . . . 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	syl6bbr 278	. . . 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 271	. . 3 ⊢ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ (𝐹 ∈ (𝐶 Cn 𝐽) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅))))
61	5, 60	biadan2 674	. 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 293	1 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)))

Syntax hints:   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436   ↦ cmpt 4729  ^◡ccnv 5113   ↾ cres 5116   “ cima 5117  ‘cfv 5888  (class class class)co 6650   ↾_t crest 16081  Topctop 20698   Cn ccn 21028  Homeochmeo 21556   CovMap ccvm 31237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-cvm 31238

This theorem is referenced by:  cvmcn  31244  cvmcov  31245