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Theorem cvmliftmoi 24975
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 2438 . . 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 24950 . 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 24974 1  |-  ( ph  ->  M  =  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711    \ cdif 3319    i^i cin 3321   (/)c0 3630   ~Pcpw 3801   {csn 3816   U.cuni 4017    e. cmpt 4269   `'ccnv 4880    |` cres 4883   "cima 4884    o. ccom 4885   ` cfv 5457  (class class class)co 6084   ↾t crest 13653    Cn ccn 17293   Conccon 17479  𝑛Locally cnlly 17533    Homeo chmeo 17790   CovMap ccvm 24947
This theorem is referenced by:  cvmliftmo  24976  cvmliftphtlem  25009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-oadd 6731  df-er 6908  df-map 7023  df-en 7113  df-fin 7116  df-fi 7419  df-rest 13655  df-topgen 13672  df-top 16968  df-bases 16970  df-topon 16971  df-cld 17088  df-nei 17167  df-cn 17296  df-con 17480  df-nlly 17535  df-hmeo 17792  df-cvm 24948
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