![]() |
Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmcov | Structured version Visualization version GIF version |
Description: Property of a covering map. In order to make the covering property more manageable, we define here the set 𝑆(𝑘) of all even coverings of an open set 𝑘 in the range. Then the covering property states that every point has a neighborhood which has an even covering. (Contributed by Mario Carneiro, 13-Feb-2015.) |
Ref | Expression |
---|---|
cvmcov.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))))}) |
cvmcov.2 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
cvmcov | ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃 ∈ 𝑋) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvmcov.1 | . . . . 5 ⊢ 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))))}) | |
2 | cvmcov.2 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 1, 2 | iscvm 34805 | . . . 4 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) ↔ ((𝐶 ∈ Top ∧ 𝐽 ∈ Top ∧ 𝐹 ∈ (𝐶 Cn 𝐽)) ∧ ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅))) |
4 | 3 | simprbi 496 | . . 3 ⊢ (𝐹 ∈ (𝐶 CovMap 𝐽) → ∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)) |
5 | eleq1 2816 | . . . . . 6 ⊢ (𝑥 = 𝑃 → (𝑥 ∈ 𝑘 ↔ 𝑃 ∈ 𝑘)) | |
6 | 5 | anbi1d 629 | . . . . 5 ⊢ (𝑥 = 𝑃 → ((𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅) ↔ (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅))) |
7 | 6 | rexbidv 3173 | . . . 4 ⊢ (𝑥 = 𝑃 → (∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅) ↔ ∃𝑘 ∈ 𝐽 (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅))) |
8 | 7 | rspcv 3603 | . . 3 ⊢ (𝑃 ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ∃𝑘 ∈ 𝐽 (𝑥 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅) → ∃𝑘 ∈ 𝐽 (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅))) |
9 | 4, 8 | mpan9 506 | . 2 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃 ∈ 𝑋) → ∃𝑘 ∈ 𝐽 (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)) |
10 | nfv 1910 | . . . 4 ⊢ Ⅎ𝑘 𝑃 ∈ 𝑥 | |
11 | nfmpt1 5250 | . . . . . . 7 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) | |
12 | 1, 11 | nfcxfr 2896 | . . . . . 6 ⊢ Ⅎ𝑘𝑆 |
13 | nfcv 2898 | . . . . . 6 ⊢ Ⅎ𝑘𝑥 | |
14 | 12, 13 | nffv 6901 | . . . . 5 ⊢ Ⅎ𝑘(𝑆‘𝑥) |
15 | nfcv 2898 | . . . . 5 ⊢ Ⅎ𝑘∅ | |
16 | 14, 15 | nfne 3038 | . . . 4 ⊢ Ⅎ𝑘(𝑆‘𝑥) ≠ ∅ |
17 | 10, 16 | nfan 1895 | . . 3 ⊢ Ⅎ𝑘(𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅) |
18 | nfv 1910 | . . 3 ⊢ Ⅎ𝑥(𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅) | |
19 | eleq2w 2812 | . . . 4 ⊢ (𝑥 = 𝑘 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑘)) | |
20 | fveq2 6891 | . . . . 5 ⊢ (𝑥 = 𝑘 → (𝑆‘𝑥) = (𝑆‘𝑘)) | |
21 | 20 | neeq1d 2995 | . . . 4 ⊢ (𝑥 = 𝑘 → ((𝑆‘𝑥) ≠ ∅ ↔ (𝑆‘𝑘) ≠ ∅)) |
22 | 19, 21 | anbi12d 630 | . . 3 ⊢ (𝑥 = 𝑘 → ((𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅) ↔ (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅))) |
23 | 17, 18, 22 | cbvrexw 3299 | . 2 ⊢ (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅) ↔ ∃𝑘 ∈ 𝐽 (𝑃 ∈ 𝑘 ∧ (𝑆‘𝑘) ≠ ∅)) |
24 | 9, 23 | sylibr 233 | 1 ⊢ ((𝐹 ∈ (𝐶 CovMap 𝐽) ∧ 𝑃 ∈ 𝑋) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑆‘𝑥) ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∀wral 3056 ∃wrex 3065 {crab 3427 ∖ cdif 3941 ∩ cin 3943 ∅c0 4318 𝒫 cpw 4598 {csn 4624 ∪ cuni 4903 ↦ cmpt 5225 ◡ccnv 5671 ↾ cres 5674 “ cima 5675 ‘cfv 6542 (class class class)co 7414 ↾t crest 17393 Topctop 22782 Cn ccn 23115 Homeochmeo 23644 CovMap ccvm 34801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-cvm 34802 |
This theorem is referenced by: cvmcov2 34821 cvmopnlem 34824 cvmfolem 34825 cvmliftmolem2 34828 cvmliftlem15 34844 cvmlift2lem10 34858 cvmlift3lem8 34872 |
Copyright terms: Public domain | W3C validator |