Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmliftmoi | Structured version Visualization version GIF version |
Description: A lift of a continuous function from a connected and locally connected space over a covering map is unique when it exists. (Contributed by Mario Carneiro, 10-Mar-2015.) |
Ref | Expression |
---|---|
cvmliftmo.b | ⊢ 𝐵 = ∪ 𝐶 |
cvmliftmo.y | ⊢ 𝑌 = ∪ 𝐾 |
cvmliftmo.f | ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
cvmliftmo.k | ⊢ (𝜑 → 𝐾 ∈ Conn) |
cvmliftmo.l | ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn) |
cvmliftmo.o | ⊢ (𝜑 → 𝑂 ∈ 𝑌) |
cvmliftmoi.m | ⊢ (𝜑 → 𝑀 ∈ (𝐾 Cn 𝐶)) |
cvmliftmoi.n | ⊢ (𝜑 → 𝑁 ∈ (𝐾 Cn 𝐶)) |
cvmliftmoi.g | ⊢ (𝜑 → (𝐹 ∘ 𝑀) = (𝐹 ∘ 𝑁)) |
cvmliftmoi.p | ⊢ (𝜑 → (𝑀‘𝑂) = (𝑁‘𝑂)) |
Ref | Expression |
---|---|
cvmliftmoi | ⊢ (𝜑 → 𝑀 = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvmliftmo.b | . 2 ⊢ 𝐵 = ∪ 𝐶 | |
2 | cvmliftmo.y | . 2 ⊢ 𝑌 = ∪ 𝐾 | |
3 | cvmliftmo.f | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) | |
4 | cvmliftmo.k | . 2 ⊢ (𝜑 → 𝐾 ∈ Conn) | |
5 | cvmliftmo.l | . 2 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn) | |
6 | cvmliftmo.o | . 2 ⊢ (𝜑 → 𝑂 ∈ 𝑌) | |
7 | cvmliftmoi.m | . 2 ⊢ (𝜑 → 𝑀 ∈ (𝐾 Cn 𝐶)) | |
8 | cvmliftmoi.n | . 2 ⊢ (𝜑 → 𝑁 ∈ (𝐾 Cn 𝐶)) | |
9 | cvmliftmoi.g | . 2 ⊢ (𝜑 → (𝐹 ∘ 𝑀) = (𝐹 ∘ 𝑁)) | |
10 | cvmliftmoi.p | . 2 ⊢ (𝜑 → (𝑀‘𝑂) = (𝑁‘𝑂)) | |
11 | eqid 2738 | . . 3 ⊢ (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) = (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 | 11 | cvmscbv 32791 | . 2 ⊢ (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑢 ∈ 𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢 ∩ 𝑣) = ∅ ∧ (𝐹 ↾ 𝑢) ∈ ((𝐶 ↾t 𝑢)Homeo(𝐽 ↾t 𝑘))))}) = (𝑏 ∈ 𝐽 ↦ {𝑚 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑚 = (◡𝐹 “ 𝑏) ∧ ∀𝑟 ∈ 𝑚 (∀𝑤 ∈ (𝑚 ∖ {𝑟})(𝑟 ∩ 𝑤) = ∅ ∧ (𝐹 ↾ 𝑟) ∈ ((𝐶 ↾t 𝑟)Homeo(𝐽 ↾t 𝑏))))}) |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12 | cvmliftmolem2 32815 | 1 ⊢ (𝜑 → 𝑀 = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∀wral 3053 {crab 3057 ∖ cdif 3840 ∩ cin 3842 ∅c0 4211 𝒫 cpw 4488 {csn 4516 ∪ cuni 4796 ↦ cmpt 5110 ◡ccnv 5524 ↾ cres 5527 “ cima 5528 ∘ ccom 5529 ‘cfv 6339 (class class class)co 7170 ↾t crest 16797 Cn ccn 21975 Conncconn 22162 𝑛-Locally cnlly 22216 Homeochmeo 22504 CovMap ccvm 32788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-map 8439 df-en 8556 df-fin 8559 df-fi 8948 df-rest 16799 df-topgen 16820 df-top 21645 df-topon 21662 df-bases 21697 df-cld 21770 df-nei 21849 df-cn 21978 df-conn 22163 df-nlly 22218 df-hmeo 22506 df-cvm 32789 |
This theorem is referenced by: cvmliftmo 32817 cvmliftphtlem 32850 |
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