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Theorem cvmliftmoi 27317
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 2454 . . 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 27292 . 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 27316 1  |-  ( ph  ->  M  =  N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   {crab 2803    \ cdif 3434    i^i cin 3436   (/)c0 3746   ~Pcpw 3969   {csn 3986   U.cuni 4200    |-> cmpt 4459   `'ccnv 4948    |` cres 4951   "cima 4952    o. ccom 4953   ` cfv 5527  (class class class)co 6201   ↾t crest 14479    Cn ccn 18961   Conccon 19148  𝑛Locally cnlly 19202   Homeochmeo 19459   CovMap ccvm 27289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-oadd 7035  df-er 7212  df-map 7327  df-en 7422  df-fin 7425  df-fi 7773  df-rest 14481  df-topgen 14502  df-top 18636  df-bases 18638  df-topon 18639  df-cld 18756  df-nei 18835  df-cn 18964  df-con 19149  df-nlly 19204  df-hmeo 19461  df-cvm 27290
This theorem is referenced by:  cvmliftmo  27318  cvmliftphtlem  27351
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