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Theorem iscvm 30976
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 680 . 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 1038 . . 3 ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ 𝐹 ∈ (𝐶 Cn 𝐽)))
32anbi1i 730 . 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 30973 . . . 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 6836 . . 3 (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝐶 ∈ Top ∧ 𝐽 ∈ Top))
6 oveq12 6619 . . . . . . 7 ((𝑐 = 𝐶𝑗 = 𝐽) → (𝑐 Cn 𝑗) = (𝐶 Cn 𝐽))
7 simpr 477 . . . . . . . . . 10 ((𝑐 = 𝐶𝑗 = 𝐽) → 𝑗 = 𝐽)
87unieqd 4417 . . . . . . . . 9 ((𝑐 = 𝐶𝑗 = 𝐽) → 𝑗 = 𝐽)
9 iscvm.2 . . . . . . . . 9 𝑋 = 𝐽
108, 9syl6eqr 2673 . . . . . . . 8 ((𝑐 = 𝐶𝑗 = 𝐽) → 𝑗 = 𝑋)
11 simpl 473 . . . . . . . . . . . . 13 ((𝑐 = 𝐶𝑗 = 𝐽) → 𝑐 = 𝐶)
1211pweqd 4140 . . . . . . . . . . . 12 ((𝑐 = 𝐶𝑗 = 𝐽) → 𝒫 𝑐 = 𝒫 𝐶)
1312difeq1d 3710 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑗 = 𝐽) → (𝒫 𝑐 ∖ {∅}) = (𝒫 𝐶 ∖ {∅}))
14 oveq1 6617 . . . . . . . . . . . . . . . 16 (𝑐 = 𝐶 → (𝑐t 𝑢) = (𝐶t 𝑢))
15 oveq1 6617 . . . . . . . . . . . . . . . 16 (𝑗 = 𝐽 → (𝑗t 𝑘) = (𝐽t 𝑘))
1614, 15oveqan12d 6629 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐶𝑗 = 𝐽) → ((𝑐t 𝑢)Homeo(𝑗t 𝑘)) = ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))
1716eleq2d 2684 . . . . . . . . . . . . . 14 ((𝑐 = 𝐶𝑗 = 𝐽) → ((𝑓𝑢) ∈ ((𝑐t 𝑢)Homeo(𝑗t 𝑘)) ↔ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))
1817anbi2d 739 . . . . . . . . . . . . 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 2981 . . . . . . . . . . . 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 739 . . . . . . . . . . 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 3145 . . . . . . . . . 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 739 . . . . . . . . 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 3145 . . . . . . . 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 3144 . . . . . . 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 3184 . . . . . 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 6638 . . . . . . 7 (𝐶 Cn 𝐽) ∈ V
2726rabex 4778 . . . . . 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 6751 . . . . 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 2684 . . . 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 4815 . . . . . . . . . . . . . 14 (𝐶 ∈ Top → 𝒫 𝐶 ∈ V)
3231adantr 481 . . . . . . . . . . . . 13 ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → 𝒫 𝐶 ∈ V)
33 difexg 4773 . . . . . . . . . . . . 13 (𝒫 𝐶 ∈ V → (𝒫 𝐶 ∖ {∅}) ∈ V)
34 rabexg 4777 . . . . . . . . . . . . 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 6253 . . . . . . . . . . . 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 495 . . . . . . . . . . 11 (((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ 𝑘𝐽) → (𝑆𝑘) = {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
3938neeq1d 2849 . . . . . . . . . 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 3937 . . . . . . . . . 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 739 . . . . . . . 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 3043 . . . . . . 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 2981 . . . . . 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 739 . . . . 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 5261 . . . . . . . . . . . . 13 (𝑓 = 𝐹𝑓 = 𝐹)
4746imaeq1d 5429 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (𝑓𝑘) = (𝐹𝑘))
4847eqeq2d 2631 . . . . . . . . . . 11 (𝑓 = 𝐹 → ( 𝑠 = (𝑓𝑘) ↔ 𝑠 = (𝐹𝑘)))
49 reseq1 5355 . . . . . . . . . . . . . 14 (𝑓 = 𝐹 → (𝑓𝑢) = (𝐹𝑢))
5049eleq1d 2683 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → ((𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)) ↔ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))
5150anbi2d 739 . . . . . . . . . . . 12 (𝑓 = 𝐹 → ((∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))) ↔ (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘)))))
5251ralbidv 2981 . . . . . . . . . . 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 746 . . . . . . . . . 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 3046 . . . . . . . . 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 739 . . . . . . . 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 3046 . . . . . . 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 2981 . . . . . 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 3350 . . . . 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 673 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  Vcvv 3189  cdif 3556  cin 3558  c0 3896  𝒫 cpw 4135  {csn 4153   cuni 4407  cmpt 4678  ccnv 5078  cres 5081  cima 5082  cfv 5852  (class class class)co 6610  t crest 16009  Topctop 20626   Cn ccn 20947  Homeochmeo 21475   CovMap ccvm 30972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-cvm 30973
This theorem is referenced by:  cvmcn  30979  cvmcov  30980
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