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Theorem cvmliftmoi 32816
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 32791 . 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 32815 1 (𝜑𝑀 = 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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