Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmliftmo Structured version   Unicode version

Theorem cvmliftmo 24972
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.) (Revised by NM, 17-Jun-2017.)
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 )
cvmliftmo.g  |-  ( ph  ->  G  e.  ( K  Cn  J ) )
cvmliftmo.p  |-  ( ph  ->  P  e.  B )
cvmliftmo.e  |-  ( ph  ->  ( F `  P
)  =  ( G `
 O ) )
Assertion
Ref Expression
cvmliftmo  |-  ( ph  ->  E* f  e.  ( K  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `  O )  =  P ) )
Distinct variable groups:    C, f    f, G    f, K    f, O    ph, f    f, F    P, f
Allowed substitution hints:    B( f)    J( f)    Y( f)

Proof of Theorem cvmliftmo
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 cvmliftmo.b . . . . 5  |-  B  = 
U. C
2 cvmliftmo.y . . . . 5  |-  Y  = 
U. K
3 cvmliftmo.f . . . . . 6  |-  ( ph  ->  F  e.  ( C CovMap  J ) )
43ad2antrr 708 . . . . 5  |-  ( ( ( ph  /\  (
f  e.  ( K  Cn  C )  /\  g  e.  ( K  Cn  C ) ) )  /\  ( ( ( F  o.  f )  =  G  /\  (
f `  O )  =  P )  /\  (
( F  o.  g
)  =  G  /\  ( g `  O
)  =  P ) ) )  ->  F  e.  ( C CovMap  J ) )
5 cvmliftmo.k . . . . . 6  |-  ( ph  ->  K  e.  Con )
65ad2antrr 708 . . . . 5  |-  ( ( ( ph  /\  (
f  e.  ( K  Cn  C )  /\  g  e.  ( K  Cn  C ) ) )  /\  ( ( ( F  o.  f )  =  G  /\  (
f `  O )  =  P )  /\  (
( F  o.  g
)  =  G  /\  ( g `  O
)  =  P ) ) )  ->  K  e.  Con )
7 cvmliftmo.l . . . . . 6  |-  ( ph  ->  K  e. 𝑛Locally  Con )
87ad2antrr 708 . . . . 5  |-  ( ( ( ph  /\  (
f  e.  ( K  Cn  C )  /\  g  e.  ( K  Cn  C ) ) )  /\  ( ( ( F  o.  f )  =  G  /\  (
f `  O )  =  P )  /\  (
( F  o.  g
)  =  G  /\  ( g `  O
)  =  P ) ) )  ->  K  e. 𝑛Locally  Con )
9 cvmliftmo.o . . . . . 6  |-  ( ph  ->  O  e.  Y )
109ad2antrr 708 . . . . 5  |-  ( ( ( ph  /\  (
f  e.  ( K  Cn  C )  /\  g  e.  ( K  Cn  C ) ) )  /\  ( ( ( F  o.  f )  =  G  /\  (
f `  O )  =  P )  /\  (
( F  o.  g
)  =  G  /\  ( g `  O
)  =  P ) ) )  ->  O  e.  Y )
11 simplrl 738 . . . . 5  |-  ( ( ( ph  /\  (
f  e.  ( K  Cn  C )  /\  g  e.  ( K  Cn  C ) ) )  /\  ( ( ( F  o.  f )  =  G  /\  (
f `  O )  =  P )  /\  (
( F  o.  g
)  =  G  /\  ( g `  O
)  =  P ) ) )  ->  f  e.  ( K  Cn  C
) )
12 simplrr 739 . . . . 5  |-  ( ( ( ph  /\  (
f  e.  ( K  Cn  C )  /\  g  e.  ( K  Cn  C ) ) )  /\  ( ( ( F  o.  f )  =  G  /\  (
f `  O )  =  P )  /\  (
( F  o.  g
)  =  G  /\  ( g `  O
)  =  P ) ) )  ->  g  e.  ( K  Cn  C
) )
13 simprll 740 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  ( K  Cn  C )  /\  g  e.  ( K  Cn  C ) ) )  /\  ( ( ( F  o.  f )  =  G  /\  (
f `  O )  =  P )  /\  (
( F  o.  g
)  =  G  /\  ( g `  O
)  =  P ) ) )  ->  ( F  o.  f )  =  G )
14 simprrl 742 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  ( K  Cn  C )  /\  g  e.  ( K  Cn  C ) ) )  /\  ( ( ( F  o.  f )  =  G  /\  (
f `  O )  =  P )  /\  (
( F  o.  g
)  =  G  /\  ( g `  O
)  =  P ) ) )  ->  ( F  o.  g )  =  G )
1513, 14eqtr4d 2472 . . . . 5  |-  ( ( ( ph  /\  (
f  e.  ( K  Cn  C )  /\  g  e.  ( K  Cn  C ) ) )  /\  ( ( ( F  o.  f )  =  G  /\  (
f `  O )  =  P )  /\  (
( F  o.  g
)  =  G  /\  ( g `  O
)  =  P ) ) )  ->  ( F  o.  f )  =  ( F  o.  g ) )
16 simprlr 741 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  ( K  Cn  C )  /\  g  e.  ( K  Cn  C ) ) )  /\  ( ( ( F  o.  f )  =  G  /\  (
f `  O )  =  P )  /\  (
( F  o.  g
)  =  G  /\  ( g `  O
)  =  P ) ) )  ->  (
f `  O )  =  P )
17 simprrr 743 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  ( K  Cn  C )  /\  g  e.  ( K  Cn  C ) ) )  /\  ( ( ( F  o.  f )  =  G  /\  (
f `  O )  =  P )  /\  (
( F  o.  g
)  =  G  /\  ( g `  O
)  =  P ) ) )  ->  (
g `  O )  =  P )
1816, 17eqtr4d 2472 . . . . 5  |-  ( ( ( ph  /\  (
f  e.  ( K  Cn  C )  /\  g  e.  ( K  Cn  C ) ) )  /\  ( ( ( F  o.  f )  =  G  /\  (
f `  O )  =  P )  /\  (
( F  o.  g
)  =  G  /\  ( g `  O
)  =  P ) ) )  ->  (
f `  O )  =  ( g `  O ) )
191, 2, 4, 6, 8, 10, 11, 12, 15, 18cvmliftmoi 24971 . . . 4  |-  ( ( ( ph  /\  (
f  e.  ( K  Cn  C )  /\  g  e.  ( K  Cn  C ) ) )  /\  ( ( ( F  o.  f )  =  G  /\  (
f `  O )  =  P )  /\  (
( F  o.  g
)  =  G  /\  ( g `  O
)  =  P ) ) )  ->  f  =  g )
2019ex 425 . . 3  |-  ( (
ph  /\  ( f  e.  ( K  Cn  C
)  /\  g  e.  ( K  Cn  C
) ) )  -> 
( ( ( ( F  o.  f )  =  G  /\  (
f `  O )  =  P )  /\  (
( F  o.  g
)  =  G  /\  ( g `  O
)  =  P ) )  ->  f  =  g ) )
2120ralrimivva 2799 . 2  |-  ( ph  ->  A. f  e.  ( K  Cn  C ) A. g  e.  ( K  Cn  C ) ( ( ( ( F  o.  f )  =  G  /\  (
f `  O )  =  P )  /\  (
( F  o.  g
)  =  G  /\  ( g `  O
)  =  P ) )  ->  f  =  g ) )
22 coeq2 5032 . . . . 5  |-  ( f  =  g  ->  ( F  o.  f )  =  ( F  o.  g ) )
2322eqeq1d 2445 . . . 4  |-  ( f  =  g  ->  (
( F  o.  f
)  =  G  <->  ( F  o.  g )  =  G ) )
24 fveq1 5728 . . . . 5  |-  ( f  =  g  ->  (
f `  O )  =  ( g `  O ) )
2524eqeq1d 2445 . . . 4  |-  ( f  =  g  ->  (
( f `  O
)  =  P  <->  ( g `  O )  =  P ) )
2623, 25anbi12d 693 . . 3  |-  ( f  =  g  ->  (
( ( F  o.  f )  =  G  /\  ( f `  O )  =  P )  <->  ( ( F  o.  g )  =  G  /\  ( g `
 O )  =  P ) ) )
2726rmo4 3128 . 2  |-  ( E* f  e.  ( K  Cn  C ) ( ( F  o.  f
)  =  G  /\  ( f `  O
)  =  P )  <->  A. f  e.  ( K  Cn  C ) A. g  e.  ( K  Cn  C ) ( ( ( ( F  o.  f )  =  G  /\  ( f `  O )  =  P )  /\  ( ( F  o.  g )  =  G  /\  (
g `  O )  =  P ) )  -> 
f  =  g ) )
2821, 27sylibr 205 1  |-  ( ph  ->  E* f  e.  ( K  Cn  C ) ( ( F  o.  f )  =  G  /\  ( f `  O )  =  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706   E*wrmo 2709   U.cuni 4016    o. ccom 4883   ` cfv 5455  (class class class)co 6082    Cn ccn 17289   Conccon 17475  𝑛Locally cnlly 17529   CovMap ccvm 24943
This theorem is referenced by:  cvmliftlem14  24985  cvmlift2lem13  25003  cvmlift3  25016
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-oadd 6729  df-er 6906  df-map 7021  df-en 7111  df-fin 7114  df-fi 7417  df-rest 13651  df-topgen 13668  df-top 16964  df-bases 16966  df-topon 16967  df-cld 17084  df-nei 17163  df-cn 17292  df-con 17476  df-nlly 17531  df-hmeo 17788  df-cvm 24944
  Copyright terms: Public domain W3C validator