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 33220 | . 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 33244 | 1 ⊢ (𝜑 → 𝑀 = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 {crab 3068 ∖ cdif 3884 ∩ cin 3886 ∅c0 4256 𝒫 cpw 4533 {csn 4561 ∪ cuni 4839 ↦ cmpt 5157 ◡ccnv 5588 ↾ cres 5591 “ cima 5592 ∘ ccom 5593 ‘cfv 6433 (class class class)co 7275 ↾t crest 17131 Cn ccn 22375 Conncconn 22562 𝑛-Locally cnlly 22616 Homeochmeo 22904 CovMap ccvm 33217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-map 8617 df-en 8734 df-fin 8737 df-fi 9170 df-rest 17133 df-topgen 17154 df-top 22043 df-topon 22060 df-bases 22096 df-cld 22170 df-nei 22249 df-cn 22378 df-conn 22563 df-nlly 22618 df-hmeo 22906 df-cvm 33218 |
This theorem is referenced by: cvmliftmo 33246 cvmliftphtlem 33279 |
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