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Theorem cvmliftmoi 33245
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.)
Hypotheses
Ref Expression
cvmliftmo.b 𝐵 = 𝐶
cvmliftmo.y 𝑌 = 𝐾
cvmliftmo.f (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
cvmliftmo.k (𝜑𝐾 ∈ Conn)
cvmliftmo.l (𝜑𝐾 ∈ 𝑛-Locally Conn)
cvmliftmo.o (𝜑𝑂𝑌)
cvmliftmoi.m (𝜑𝑀 ∈ (𝐾 Cn 𝐶))
cvmliftmoi.n (𝜑𝑁 ∈ (𝐾 Cn 𝐶))
cvmliftmoi.g (𝜑 → (𝐹𝑀) = (𝐹𝑁))
cvmliftmoi.p (𝜑 → (𝑀𝑂) = (𝑁𝑂))
Assertion
Ref Expression
cvmliftmoi (𝜑𝑀 = 𝑁)

Proof of Theorem cvmliftmoi
Dummy variables 𝑏 𝑘 𝑚 𝑟 𝑠 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvmliftmo.b . 2 𝐵 = 𝐶
2 cvmliftmo.y . 2 𝑌 = 𝐾
3 cvmliftmo.f . 2 (𝜑𝐹 ∈ (𝐶 CovMap 𝐽))
4 cvmliftmo.k . 2 (𝜑𝐾 ∈ Conn)
5 cvmliftmo.l . 2 (𝜑𝐾 ∈ 𝑛-Locally Conn)
6 cvmliftmo.o . 2 (𝜑𝑂𝑌)
7 cvmliftmoi.m . 2 (𝜑𝑀 ∈ (𝐾 Cn 𝐶))
8 cvmliftmoi.n . 2 (𝜑𝑁 ∈ (𝐾 Cn 𝐶))
9 cvmliftmoi.g . 2 (𝜑 → (𝐹𝑀) = (𝐹𝑁))
10 cvmliftmoi.p . 2 (𝜑 → (𝑀𝑂) = (𝑁𝑂))
11 eqid 2738 . . 3 (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))}) = (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))})
1211cvmscbv 33220 . 2 (𝑘𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑠 = (𝐹𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝐹𝑢) ∈ ((𝐶t 𝑢)Homeo(𝐽t 𝑘))))}) = (𝑏𝐽 ↦ {𝑚 ∈ (𝒫 𝐶 ∖ {∅}) ∣ ( 𝑚 = (𝐹𝑏) ∧ ∀𝑟𝑚 (∀𝑤 ∈ (𝑚 ∖ {𝑟})(𝑟𝑤) = ∅ ∧ (𝐹𝑟) ∈ ((𝐶t 𝑟)Homeo(𝐽t 𝑏))))})
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12cvmliftmolem2 33244 1 (𝜑𝑀 = 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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