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Theorem cvmcov 23809
Description: Property of a covering map. In order to make the covering property more manageable, we define here the set  S ( k ) of all even coverings of an open set  k in the range. Then the covering property states that every point has a neighborhood which has an even covering. (Contributed by Mario Carneiro, 13-Feb-2015.)
Hypotheses
Ref Expression
cvmcov.1  |-  S  =  ( 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 ) ) ) ) } )
cvmcov.2  |-  X  = 
U. J
Assertion
Ref Expression
cvmcov  |-  ( ( F  e.  ( C CovMap  J )  /\  P  e.  X )  ->  E. x  e.  J  ( P  e.  x  /\  ( S `  x )  =/=  (/) ) )
Distinct variable groups:    k, s, u, v, x, C    k, F, s, u, v, x    P, k, x    k, J, s, u, v, x   
x, S    x, X
Allowed substitution hints:    P( v, u, s)    S( v, u, k, s)    X( v, u, k, s)

Proof of Theorem cvmcov
StepHypRef Expression
1 cvmcov.1 . . . . 5  |-  S  =  ( 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 ) ) ) ) } )
2 cvmcov.2 . . . . 5  |-  X  = 
U. J
31, 2iscvm 23805 . . . 4  |-  ( F  e.  ( C CovMap  J
)  <->  ( ( C  e.  Top  /\  J  e.  Top  /\  F  e.  ( C  Cn  J
) )  /\  A. x  e.  X  E. k  e.  J  (
x  e.  k  /\  ( S `  k )  =/=  (/) ) ) )
43simprbi 450 . . 3  |-  ( F  e.  ( C CovMap  J
)  ->  A. x  e.  X  E. k  e.  J  ( x  e.  k  /\  ( S `  k )  =/=  (/) ) )
5 eleq1 2356 . . . . . 6  |-  ( x  =  P  ->  (
x  e.  k  <->  P  e.  k ) )
65anbi1d 685 . . . . 5  |-  ( x  =  P  ->  (
( x  e.  k  /\  ( S `  k )  =/=  (/) )  <->  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) ) )
76rexbidv 2577 . . . 4  |-  ( x  =  P  ->  ( E. k  e.  J  ( x  e.  k  /\  ( S `  k
)  =/=  (/) )  <->  E. k  e.  J  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) ) )
87rspcv 2893 . . 3  |-  ( P  e.  X  ->  ( A. x  e.  X  E. k  e.  J  ( x  e.  k  /\  ( S `  k
)  =/=  (/) )  ->  E. k  e.  J  ( P  e.  k  /\  ( S `  k
)  =/=  (/) ) ) )
94, 8mpan9 455 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  P  e.  X )  ->  E. k  e.  J  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) )
10 nfv 1609 . . . 4  |-  F/ k  P  e.  x
11 nfmpt1 4125 . . . . . . 7  |-  F/_ 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 ) ) ) ) } )
121, 11nfcxfr 2429 . . . . . 6  |-  F/_ k S
13 nfcv 2432 . . . . . 6  |-  F/_ k
x
1412, 13nffv 5548 . . . . 5  |-  F/_ k
( S `  x
)
15 nfcv 2432 . . . . 5  |-  F/_ k (/)
1614, 15nfne 2552 . . . 4  |-  F/ k ( S `  x
)  =/=  (/)
1710, 16nfan 1783 . . 3  |-  F/ k ( P  e.  x  /\  ( S `  x
)  =/=  (/) )
18 nfv 1609 . . 3  |-  F/ x
( P  e.  k  /\  ( S `  k )  =/=  (/) )
19 eleq2 2357 . . . 4  |-  ( x  =  k  ->  ( P  e.  x  <->  P  e.  k ) )
20 fveq2 5541 . . . . 5  |-  ( x  =  k  ->  ( S `  x )  =  ( S `  k ) )
2120neeq1d 2472 . . . 4  |-  ( x  =  k  ->  (
( S `  x
)  =/=  (/)  <->  ( S `  k )  =/=  (/) ) )
2219, 21anbi12d 691 . . 3  |-  ( x  =  k  ->  (
( P  e.  x  /\  ( S `  x
)  =/=  (/) )  <->  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) ) )
2317, 18, 22cbvrex 2774 . 2  |-  ( E. x  e.  J  ( P  e.  x  /\  ( S `  x )  =/=  (/) )  <->  E. k  e.  J  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) )
249, 23sylibr 203 1  |-  ( ( F  e.  ( C CovMap  J )  /\  P  e.  X )  ->  E. x  e.  J  ( P  e.  x  /\  ( S `  x )  =/=  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    \ cdif 3162    i^i cin 3164   (/)c0 3468   ~Pcpw 3638   {csn 3653   U.cuni 3843    e. cmpt 4093   `'ccnv 4704    |` cres 4707   "cima 4708   ` cfv 5271  (class class class)co 5874   ↾t crest 13341   Topctop 16647    Cn ccn 16970    Homeo chmeo 17460   CovMap ccvm 23801
This theorem is referenced by:  cvmcov2  23821  cvmopnlem  23824  cvmfolem  23825  cvmliftmolem2  23828  cvmliftlem15  23844  cvmlift2lem10  23858  cvmlift3lem8  23872
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-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  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-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-cvm 23802
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