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Theorem cvmliftmoi 28957
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  = 
U. C
cvmliftmo.y  |-  Y  = 
U. K
cvmliftmo.f  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
cvmliftmo.k  |-  ( ph  ->  K  e.  Con )
cvmliftmo.l  |-  ( ph  ->  K  e. 𝑛Locally  Con )
cvmliftmo.o  |-  ( ph  ->  O  e.  Y )
cvmliftmoi.m  |-  ( ph  ->  M  e.  ( K  Cn  C ) )
cvmliftmoi.n  |-  ( ph  ->  N  e.  ( K  Cn  C ) )
cvmliftmoi.g  |-  ( ph  ->  ( F  o.  M
)  =  ( F  o.  N ) )
cvmliftmoi.p  |-  ( ph  ->  ( M `  O
)  =  ( N `
 O ) )
Assertion
Ref Expression
cvmliftmoi  |-  ( ph  ->  M  =  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  = 
U. C
2 cvmliftmo.y . 2  |-  Y  = 
U. K
3 cvmliftmo.f . 2  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
4 cvmliftmo.k . 2  |-  ( ph  ->  K  e.  Con )
5 cvmliftmo.l . 2  |-  ( ph  ->  K  e. 𝑛Locally  Con )
6 cvmliftmo.o . 2  |-  ( ph  ->  O  e.  Y )
7 cvmliftmoi.m . 2  |-  ( ph  ->  M  e.  ( K  Cn  C ) )
8 cvmliftmoi.n . 2  |-  ( ph  ->  N  e.  ( K  Cn  C ) )
9 cvmliftmoi.g . 2  |-  ( ph  ->  ( F  o.  M
)  =  ( F  o.  N ) )
10 cvmliftmoi.p . 2  |-  ( ph  ->  ( M `  O
)  =  ( N `
 O ) )
11 eqid 2396 . . 3  |-  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
k )  /\  A. u  e.  s  ( A. v  e.  (
s  \  { u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u ) Homeo ( Jt  k ) ) ) ) } )  =  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F " k )  /\  A. u  e.  s  ( A. v  e.  ( s  \  { u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u ) Homeo ( Jt  k ) ) ) ) } )
1211cvmscbv 28932 . 2  |-  ( k  e.  J  |->  { s  e.  ( ~P C  \  { (/) } )  |  ( U. s  =  ( `' F "
k )  /\  A. u  e.  s  ( A. v  e.  (
s  \  { u } ) ( u  i^i  v )  =  (/)  /\  ( F  |`  u )  e.  ( ( Ct  u ) Homeo ( Jt  k ) ) ) ) } )  =  ( b  e.  J  |->  { m  e.  ( ~P C  \  { (/) } )  |  ( U. m  =  ( `' F " b )  /\  A. r  e.  m  ( A. w  e.  ( m  \  { r } ) ( r  i^i  w )  =  (/)  /\  ( F  |`  r )  e.  ( ( Ct  r ) Homeo ( Jt  b ) ) ) ) } )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12cvmliftmolem2 28956 1  |-  ( ph  ->  M  =  N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836   A.wral 2746   {crab 2750    \ cdif 3403    i^i cin 3405   (/)c0 3728   ~Pcpw 3944   {csn 3961   U.cuni 4180    |-> cmpt 4442   `'ccnv 4929    |` cres 4932   "cima 4933    o. ccom 4934   ` cfv 5513  (class class class)co 6218   ↾t crest 14851    Cn ccn 19834   Conccon 20020  𝑛Locally cnlly 20074   Homeochmeo 20362   CovMap ccvm 28929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rmo 2754  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4181  df-int 4217  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-tr 4478  df-eprel 4722  df-id 4726  df-po 4731  df-so 4732  df-fr 4769  df-we 4771  df-ord 4812  df-on 4813  df-lim 4814  df-suc 4815  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-om 6622  df-1st 6721  df-2nd 6722  df-recs 6982  df-rdg 7016  df-oadd 7074  df-er 7251  df-map 7362  df-en 7458  df-fin 7461  df-fi 7808  df-rest 14853  df-topgen 14874  df-top 19507  df-bases 19509  df-topon 19510  df-cld 19628  df-nei 19708  df-cn 19837  df-con 20021  df-nlly 20076  df-hmeo 20364  df-cvm 28930
This theorem is referenced by:  cvmliftmo  28958  cvmliftphtlem  28991
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