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Theorem cvmliftmoi 24923
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 2404 . . 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 24898 . 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 24922 1  |-  ( ph  ->  M  =  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   {crab 2670    \ cdif 3277    i^i cin 3279   (/)c0 3588   ~Pcpw 3759   {csn 3774   U.cuni 3975    e. cmpt 4226   `'ccnv 4836    |` cres 4839   "cima 4840    o. ccom 4841   ` cfv 5413  (class class class)co 6040   ↾t crest 13603    Cn ccn 17242   Conccon 17427  𝑛Locally cnlly 17481    Homeo chmeo 17738   CovMap ccvm 24895
This theorem is referenced by:  cvmliftmo  24924  cvmliftphtlem  24957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-fin 7072  df-fi 7374  df-rest 13605  df-topgen 13622  df-top 16918  df-bases 16920  df-topon 16921  df-cld 17038  df-nei 17117  df-cn 17245  df-con 17428  df-nlly 17483  df-hmeo 17740  df-cvm 24896
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