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Theorem cvmliftmoi 24218
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 2358 . . 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 24193 . 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 24217 1  |-  ( ph  ->  M  =  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   {crab 2623    \ cdif 3225    i^i cin 3227   (/)c0 3531   ~Pcpw 3701   {csn 3716   U.cuni 3908    e. cmpt 4158   `'ccnv 4770    |` cres 4773   "cima 4774    o. ccom 4775   ` cfv 5337  (class class class)co 5945   ↾t crest 13424    Cn ccn 17060   Conccon 17243  𝑛Locally cnlly 17297    Homeo chmeo 17550   CovMap ccvm 24190
This theorem is referenced by:  cvmliftmo  24219  cvmliftphtlem  24252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-oadd 6570  df-er 6747  df-map 6862  df-en 6952  df-fin 6955  df-fi 7255  df-rest 13426  df-topgen 13443  df-top 16742  df-bases 16744  df-topon 16745  df-cld 16862  df-nei 16941  df-cn 17063  df-con 17244  df-nlly 17299  df-hmeo 17552  df-cvm 24191
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