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Theorem cvmliftmo 23830
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 2331 . . . . 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 2331 . . . . 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 23829 . . . 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 2648 . 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 4858 . . . . 5  |-  ( f  =  g  ->  ( F  o.  f )  =  ( F  o.  g ) )
2322eqeq1d 2304 . . . 4  |-  ( f  =  g  ->  (
( F  o.  f
)  =  G  <->  ( F  o.  g )  =  G ) )
24 fveq1 5540 . . . . 5  |-  ( f  =  g  ->  (
f `  O )  =  ( g `  O ) )
2524eqeq1d 2304 . . . 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 2971 . 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 1632    e. wcel 1696   A.wral 2556   E*wrmo 2559   U.cuni 3843    o. ccom 4709   ` cfv 5271  (class class class)co 5874    Cn ccn 16970   Conccon 17153  𝑛Locally cnlly 17207   CovMap ccvm 23801
This theorem is referenced by:  cvmliftlem14  23843  cvmlift2lem13  23861  cvmlift3  23874
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-fin 6883  df-fi 7181  df-rest 13343  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655  df-cld 16772  df-nei 16851  df-cn 16973  df-con 17154  df-nlly 17209  df-hmeo 17462  df-cvm 23802
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