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 2818 | . . 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 32402 | . 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 32426 | 1 ⊢ (𝜑 → 𝑀 = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 {crab 3139 ∖ cdif 3930 ∩ cin 3932 ∅c0 4288 𝒫 cpw 4535 {csn 4557 ∪ cuni 4830 ↦ cmpt 5137 ◡ccnv 5547 ↾ cres 5550 “ cima 5551 ∘ ccom 5552 ‘cfv 6348 (class class class)co 7145 ↾t crest 16682 Cn ccn 21760 Conncconn 21947 𝑛-Locally cnlly 22001 Homeochmeo 22289 CovMap ccvm 32399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-fin 8501 df-fi 8863 df-rest 16684 df-topgen 16705 df-top 21430 df-topon 21447 df-bases 21482 df-cld 21555 df-nei 21634 df-cn 21763 df-conn 21948 df-nlly 22003 df-hmeo 22291 df-cvm 32400 |
This theorem is referenced by: cvmliftmo 32428 cvmliftphtlem 32461 |
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