Step | Hyp | Ref
| Expression |
1 | | anass 472 |
. 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 1091 |
. . 3
⊢ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) ∧ 𝐹 ∈ (𝐶 Cn 𝐽))) |
3 | 2 | anbi1i 627 |
. 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 32961 |
. . . 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 7468 |
. . 3
⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → (𝐶 ∈ Top ∧ 𝐽 ∈ Top)) |
6 | | oveq12 7243 |
. . . . . . 7
⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → (𝑐 Cn 𝑗) = (𝐶 Cn 𝐽)) |
7 | | simpr 488 |
. . . . . . . . . 10
⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → 𝑗 = 𝐽) |
8 | 7 | unieqd 4849 |
. . . . . . . . 9
⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = ∪
𝐽) |
9 | | iscvm.2 |
. . . . . . . . 9
⊢ 𝑋 = ∪
𝐽 |
10 | 8, 9 | eqtr4di 2798 |
. . . . . . . 8
⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋) |
11 | | simpl 486 |
. . . . . . . . . . . . 13
⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → 𝑐 = 𝐶) |
12 | 11 | pweqd 4548 |
. . . . . . . . . . . 12
⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → 𝒫 𝑐 = 𝒫 𝐶) |
13 | 12 | difeq1d 4052 |
. . . . . . . . . . 11
⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → (𝒫 𝑐 ∖ {∅}) = (𝒫 𝐶 ∖
{∅})) |
14 | | oveq1 7241 |
. . . . . . . . . . . . . . . 16
⊢ (𝑐 = 𝐶 → (𝑐 ↾t 𝑢) = (𝐶 ↾t 𝑢)) |
15 | | oveq1 7241 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝐽 → (𝑗 ↾t 𝑘) = (𝐽 ↾t 𝑘)) |
16 | 14, 15 | oveqan12d 7253 |
. . . . . . . . . . . . . . 15
⊢ ((𝑐 = 𝐶 ∧ 𝑗 = 𝐽) → ((𝑐 ↾t 𝑢)Homeo(𝑗 ↾t 𝑘)) = ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))) |
17 | 16 | eleq2d 2825 |
. . . . . . . . . . . . . 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 632 |
. . . . . . . . . . . . 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 3121 |
. . . . . . . . . . . 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 632 |
. . . . . . . . . . 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 3329 |
. . . . . . . . . 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 632 |
. . . . . . . . 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 3329 |
. . . . . . . 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 3328 |
. . . . . . 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 3411 |
. . . . . 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 7267 |
. . . . . . 7
⊢ (𝐶 Cn 𝐽) ∈ V |
27 | 26 | rabex 5241 |
. . . . . 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 7385 |
. . . . 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 2825 |
. . . 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 5287 |
. . . . . . . . . . . . . 14
⊢ (𝐶 ∈ Top → 𝒫
𝐶 ∈
V) |
32 | 31 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top) → 𝒫
𝐶 ∈
V) |
33 | | difexg 5236 |
. . . . . . . . . . . . 13
⊢
(𝒫 𝐶 ∈
V → (𝒫 𝐶
∖ {∅}) ∈ V) |
34 | | rabexg 5240 |
. . . . . . . . . . . . 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 6850 |
. . . . . . . . . . . 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 600 |
. . . . . . . . . . 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 3003 |
. . . . . . . . . 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 4316 |
. . . . . . . . . 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 290 |
. . . . . . . . 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 632 |
. . . . . . . 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 3225 |
. . . . . . 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 3121 |
. . . . . 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 632 |
. . . . 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 5759 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹) |
47 | 46 | imaeq1d 5945 |
. . . . . . . . . . . 12
⊢ (𝑓 = 𝐹 → (◡𝑓 “ 𝑘) = (◡𝐹 “ 𝑘)) |
48 | 47 | eqeq2d 2750 |
. . . . . . . . . . 11
⊢ (𝑓 = 𝐹 → (∪ 𝑠 = (◡𝑓 “ 𝑘) ↔ ∪ 𝑠 = (◡𝐹 “ 𝑘))) |
49 | | reseq1 5862 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = 𝐹 → (𝑓 ↾ 𝑢) = (𝐹 ↾ 𝑢)) |
50 | 49 | eleq1d 2824 |
. . . . . . . . . . . . 13
⊢ (𝑓 = 𝐹 → ((𝑓 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)) ↔ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘)))) |
51 | 50 | anbi2d 632 |
. . . . . . . . . . . 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 3121 |
. . . . . . . . . . 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 634 |
. . . . . . . . . 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 3226 |
. . . . . . . . 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 632 |
. . . . . . . 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 3226 |
. . . . . . 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 3121 |
. . . . . 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 3617 |
. . . . 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 292 |
. . . 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 285 |
. . 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 822 |
. 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 307 |
1
⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅))) |