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Theorem iscvm 31540
Description: The property of being a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
Hypotheses
Ref Expression
iscvm.1 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
iscvm.2 𝑋 = 𝐽
Assertion
Ref Expression
iscvm (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅)))
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.
StepHypRef Expression
1 anass 684 . 2 ((((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅)) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ (𝐹 ∈ (𝐶 Cn 𝐽) ∧ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅))))
2 df-3an 1074 . . 3 ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ 𝐹 ∈ (𝐶 Cn 𝐽)))
32anbi1i 733 . 2 (((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅)) ↔ (((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅)))
4 df-cvm 31537 . . . 4 CovMap = (𝑐 ∈ Top, 𝑗 ∈ Top ↦ {𝑓 ∈ (𝑐 Cn 𝑗) ∣ ∀𝑥 𝑗𝑘𝑗 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝑐t 𝑢)Homeo(𝑗t 𝑘)))))})
54elmpt2cl 7033 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝐶 ∈ Top ∧ 𝐽 ∈ Top))
6 oveq12 6814 . . . . . . 7 ((𝑐 = 𝐶𝑗 = 𝐽) → (𝑐 Cn 𝑗) = (𝐶 Cn 𝐽))
7 simpr 479 . . . . . . . . . 10 ((𝑐 = 𝐶𝑗 = 𝐽) → 𝑗 = 𝐽)
87unieqd 4590 . . . . . . . . 9 ((𝑐 = 𝐶𝑗 = 𝐽) → 𝑗 = 𝐽)
9 iscvm.2 . . . . . . . . 9 𝑋 = 𝐽
108, 9syl6eqr 2804 . . . . . . . 8 ((𝑐 = 𝐶𝑗 = 𝐽) → 𝑗 = 𝑋)
11 simpl 474 . . . . . . . . . . . . 13 ((𝑐 = 𝐶𝑗 = 𝐽) → 𝑐 = 𝐶)
1211pweqd 4299 . . . . . . . . . . . 12 ((𝑐 = 𝐶𝑗 = 𝐽) → 𝒫 𝑐 = 𝒫 𝐶)
1312difeq1d 3862 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑗 = 𝐽) → (𝒫 𝑐 ∖ {∅}) = (𝒫 𝐶 ∖ {∅}))
14 oveq1 6812 . . . . . . . . . . . . . . . 16 (𝑐 = 𝐶 → (𝑐t 𝑢) = (𝐶t 𝑢))
15 oveq1 6812 . . . . . . . . . . . . . . . 16 (𝑗 = 𝐽 → (𝑗t 𝑘) = (𝐽t 𝑘))
1614, 15oveqan12d 6824 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐶𝑗 = 𝐽) → ((𝑐t 𝑢)Homeo(𝑗t 𝑘)) = ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))
1716eleq2d 2817 . . . . . . . . . . . . . 14 ((𝑐 = 𝐶𝑗 = 𝐽) → ((𝑓𝑢) ∈ ((𝑐t 𝑢)Homeo(𝑗t 𝑘)) ↔ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))
1817anbi2d 742 . . . . . . . . . . . . 13 ((𝑐 = 𝐶𝑗 = 𝐽) → ((∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝑐t 𝑢)Homeo(𝑗t 𝑘))) ↔ (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))
1918ralbidv 3116 . . . . . . . . . . . 12 ((𝑐 = 𝐶𝑗 = 𝐽) → (∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝑐t 𝑢)Homeo(𝑗t 𝑘))) ↔ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))
2019anbi2d 742 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑗 = 𝐽) → (( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝑐t 𝑢)Homeo(𝑗t 𝑘)))) ↔ ( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))))
2113, 20rexeqbidv 3284 . . . . . . . . . 10 ((𝑐 = 𝐶𝑗 = 𝐽) → (∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝑐t 𝑢)Homeo(𝑗t 𝑘)))) ↔ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))))
2221anbi2d 742 . . . . . . . . 9 ((𝑐 = 𝐶𝑗 = 𝐽) → ((𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝑐t 𝑢)Homeo(𝑗t 𝑘))))) ↔ (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))))
237, 22rexeqbidv 3284 . . . . . . . 8 ((𝑐 = 𝐶𝑗 = 𝐽) → (∃𝑘𝑗 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝑐t 𝑢)Homeo(𝑗t 𝑘))))) ↔ ∃𝑘𝐽 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))))
2410, 23raleqbidv 3283 . . . . . . 7 ((𝑐 = 𝐶𝑗 = 𝐽) → (∀𝑥 𝑗𝑘𝑗 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝑐t 𝑢)Homeo(𝑗t 𝑘))))) ↔ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))))
256, 24rabeqbidv 3327 . . . . . 6 ((𝑐 = 𝐶𝑗 = 𝐽) → {𝑓 ∈ (𝑐 Cn 𝑗) ∣ ∀𝑥 𝑗𝑘𝑗 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝑐t 𝑢)Homeo(𝑗t 𝑘)))))} = {𝑓 ∈ (𝐶 Cn 𝐽) ∣ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))})
26 ovex 6833 . . . . . . 7 (𝐶 Cn 𝐽) ∈ V
2726rabex 4956 . . . . . 6 {𝑓 ∈ (𝐶 Cn 𝐽) ∣ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))} ∈ V
2825, 4, 27ovmpt2a 6948 . . . . 5 ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (𝐶 CovMap 𝐽) = {𝑓 ∈ (𝐶 Cn 𝐽) ∣ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))})
2928eleq2d 2817 . . . 4 ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ 𝐹 ∈ {𝑓 ∈ (𝐶 Cn 𝐽) ∣ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))}))
30 id 22 . . . . . . . . . . . 12 (𝑘𝐽𝑘𝐽)
31 pwexg 4991 . . . . . . . . . . . . . 14 (𝐶 ∈ Top → 𝒫 𝐶 ∈ V)
3231adantr 472 . . . . . . . . . . . . 13 ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → 𝒫 𝐶 ∈ V)
33 difexg 4952 . . . . . . . . . . . . 13 (𝒫 𝐶 ∈ V → (𝒫 𝐶 ∖ {∅}) ∈ V)
34 rabexg 4955 . . . . . . . . . . . . 13 ((𝒫 𝐶 ∖ {∅}) ∈ V → {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))} ∈ V)
3532, 33, 343syl 18 . . . . . . . . . . . 12 ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))} ∈ V)
36 iscvm.1 . . . . . . . . . . . . 13 𝑆 = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
3736fvmpt2 6445 . . . . . . . . . . . 12 ((𝑘𝐽 ∧ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))} ∈ V) → (𝑆𝑘) = {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
3830, 35, 37syl2anr 496 . . . . . . . . . . 11 (((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ 𝑘𝐽) → (𝑆𝑘) = {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
3938neeq1d 2983 . . . . . . . . . 10 (((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ 𝑘𝐽) → ((𝑆𝑘) ≠ ∅ ↔ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))} ≠ ∅))
40 rabn0 4093 . . . . . . . . . 10 ({𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))} ≠ ∅ ↔ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))
4139, 40syl6bb 276 . . . . . . . . 9 (((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ 𝑘𝐽) → ((𝑆𝑘) ≠ ∅ ↔ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))))
4241anbi2d 742 . . . . . . . 8 (((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ 𝑘𝐽) → ((𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅) ↔ (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))))
4342rexbidva 3179 . . . . . . 7 ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (∃𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅) ↔ ∃𝑘𝐽 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))))
4443ralbidv 3116 . . . . . 6 ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅) ↔ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))))
4544anbi2d 742 . . . . 5 ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → ((𝐹 ∈ (𝐶 Cn 𝐽) ∧ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅)) ↔ (𝐹 ∈ (𝐶 Cn 𝐽) ∧ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))))))
46 cnveq 5443 . . . . . . . . . . . . 13 (𝑓 = 𝐹𝑓 = 𝐹)
4746imaeq1d 5615 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (𝑓𝑘) = (𝐹𝑘))
4847eqeq2d 2762 . . . . . . . . . . 11 (𝑓 = 𝐹 → ( 𝑠 = (𝑓𝑘) ↔ 𝑠 = (𝐹𝑘)))
49 reseq1 5537 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → (𝑓𝑢) = (𝐹𝑢))
5049eleq1d 2816 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → ((𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)) ↔ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))
5150anbi2d 742 . . . . . . . . . . . 12 (𝑓 = 𝐹 → ((∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))) ↔ (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))
5251ralbidv 3116 . . . . . . . . . . 11 (𝑓 = 𝐹 → (∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))) ↔ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))
5348, 52anbi12d 749 . . . . . . . . . 10 (𝑓 = 𝐹 → (( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))) ↔ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))))
5453rexbidv 3182 . . . . . . . . 9 (𝑓 = 𝐹 → (∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))) ↔ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))))
5554anbi2d 742 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))) ↔ (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))))
5655rexbidv 3182 . . . . . . 7 (𝑓 = 𝐹 → (∃𝑘𝐽 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))) ↔ ∃𝑘𝐽 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))))
5756ralbidv 3116 . . . . . 6 (𝑓 = 𝐹 → (∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))) ↔ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))))
5857elrab 3496 . . . . 5 (𝐹 ∈ {𝑓 ∈ (𝐶 Cn 𝐽) ∣ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))} ↔ (𝐹 ∈ (𝐶 Cn 𝐽) ∧ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))))
5945, 58syl6bbr 278 . . . 4 ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → ((𝐹 ∈ (𝐶 Cn 𝐽) ∧ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅)) ↔ 𝐹 ∈ {𝑓 ∈ (𝐶 Cn 𝐽) ∣ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝐶 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))}))
6029, 59bitr4d 271 . . 3 ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ (𝐹 ∈ (𝐶 Cn 𝐽) ∧ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅))))
615, 60biadan2 677 . 2 (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ (𝐹 ∈ (𝐶 Cn 𝐽) ∧ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅))))
621, 3, 613bitr4ri 293 1 (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥𝑋𝑘𝐽 (𝑥𝑘 ∧ (𝑆𝑘) ≠ ∅)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  wne 2924  wral 3042  wrex 3043  {crab 3046  Vcvv 3332  cdif 3704  cin 3706  c0 4050  𝒫 cpw 4294  {csn 4313   cuni 4580  cmpt 4873  ccnv 5257  cres 5260  cima 5261  cfv 6041  (class class class)co 6805  t crest 16275  Topctop 20892   Cn ccn 21222  Homeochmeo 21750   CovMap ccvm 31536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-cvm 31537
This theorem is referenced by:  cvmcn  31543  cvmcov  31544
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