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Theorem cvmliftmoi 35026
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 2725 . . 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 35001 . 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 35025 1 (𝜑𝑀 = 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3050  {crab 3418  cdif 3941  cin 3943  c0 4322  𝒫 cpw 4604  {csn 4630   cuni 4909  cmpt 5232  ccnv 5677  cres 5680  cima 5681  ccom 5682  cfv 6549  (class class class)co 7419  t crest 17410   Cn ccn 23177  Conncconn 23364  𝑛-Locally cnlly 23418  Homeochmeo 23706   CovMap ccvm 34998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-map 8847  df-en 8965  df-fin 8968  df-fi 9441  df-rest 17412  df-topgen 17433  df-top 22845  df-topon 22862  df-bases 22898  df-cld 22972  df-nei 23051  df-cn 23180  df-conn 23365  df-nlly 23420  df-hmeo 23708  df-cvm 34999
This theorem is referenced by:  cvmliftmo  35027  cvmliftphtlem  35060
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