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Theorem cvmliftmo 23815
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 706 . . . . 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 706 . . . . 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 706 . . . . 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 706 . . . . 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 736 . . . . 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 737 . . . . 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 738 . . . . . 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 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.  g )  =  G )
1513, 14eqtr4d 2318 . . . . 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 739 . . . . . 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 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 ) ) )  ->  (
g `  O )  =  P )
1816, 17eqtr4d 2318 . . . . 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 23814 . . . 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 423 . . 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 2635 . 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 4842 . . . . 5  |-  ( f  =  g  ->  ( F  o.  f )  =  ( F  o.  g ) )
2322eqeq1d 2291 . . . 4  |-  ( f  =  g  ->  (
( F  o.  f
)  =  G  <->  ( F  o.  g )  =  G ) )
24 fveq1 5524 . . . . 5  |-  ( f  =  g  ->  (
f `  O )  =  ( g `  O ) )
2524eqeq1d 2291 . . . 4  |-  ( f  =  g  ->  (
( f `  O
)  =  P  <->  ( g `  O )  =  P ) )
2623, 25anbi12d 691 . . 3  |-  ( f  =  g  ->  (
( ( F  o.  f )  =  G  /\  ( f `  O )  =  P )  <->  ( ( F  o.  g )  =  G  /\  ( g `
 O )  =  P ) ) )
2726rmo4 2958 . 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 203 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 358    = wceq 1623    e. wcel 1684   A.wral 2543   E*wrmo 2546   U.cuni 3827    o. ccom 4693   ` cfv 5255  (class class class)co 5858    Cn ccn 16954   Conccon 17137  𝑛Locally cnlly 17191   CovMap ccvm 23786
This theorem is referenced by:  cvmliftlem14  23828  cvmlift2lem13  23846  cvmlift3  23859
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-fin 6867  df-fi 7165  df-rest 13327  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cld 16756  df-nei 16835  df-cn 16957  df-con 17138  df-nlly 17193  df-hmeo 17446  df-cvm 23787
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