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Theorem cvmcov 24981
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 24977 . . . 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 452 . . 3  |-  ( F  e.  ( C CovMap  J
)  ->  A. x  e.  X  E. k  e.  J  ( x  e.  k  /\  ( S `  k )  =/=  (/) ) )
5 eleq1 2502 . . . . . 6  |-  ( x  =  P  ->  (
x  e.  k  <->  P  e.  k ) )
65anbi1d 687 . . . . 5  |-  ( x  =  P  ->  (
( x  e.  k  /\  ( S `  k )  =/=  (/) )  <->  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) ) )
76rexbidv 2732 . . . 4  |-  ( x  =  P  ->  ( E. k  e.  J  ( x  e.  k  /\  ( S `  k
)  =/=  (/) )  <->  E. k  e.  J  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) ) )
87rspcv 3054 . . 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 457 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  P  e.  X )  ->  E. k  e.  J  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) )
10 nfv 1630 . . . 4  |-  F/ k  P  e.  x
11 nfmpt1 4323 . . . . . . 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 2575 . . . . . 6  |-  F/_ k S
13 nfcv 2578 . . . . . 6  |-  F/_ k
x
1412, 13nffv 5764 . . . . 5  |-  F/_ k
( S `  x
)
15 nfcv 2578 . . . . 5  |-  F/_ k (/)
1614, 15nfne 2701 . . . 4  |-  F/ k ( S `  x
)  =/=  (/)
1710, 16nfan 1848 . . 3  |-  F/ k ( P  e.  x  /\  ( S `  x
)  =/=  (/) )
18 nfv 1630 . . 3  |-  F/ x
( P  e.  k  /\  ( S `  k )  =/=  (/) )
19 eleq2 2503 . . . 4  |-  ( x  =  k  ->  ( P  e.  x  <->  P  e.  k ) )
20 fveq2 5757 . . . . 5  |-  ( x  =  k  ->  ( S `  x )  =  ( S `  k ) )
2120neeq1d 2620 . . . 4  |-  ( x  =  k  ->  (
( S `  x
)  =/=  (/)  <->  ( S `  k )  =/=  (/) ) )
2219, 21anbi12d 693 . . 3  |-  ( x  =  k  ->  (
( P  e.  x  /\  ( S `  x
)  =/=  (/) )  <->  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) ) )
2317, 18, 22cbvrex 2935 . 2  |-  ( E. x  e.  J  ( P  e.  x  /\  ( S `  x )  =/=  (/) )  <->  E. k  e.  J  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) )
249, 23sylibr 205 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 360    /\ w3a 937    = wceq 1653    e. wcel 1727    =/= wne 2605   A.wral 2711   E.wrex 2712   {crab 2715    \ cdif 3303    i^i cin 3305   (/)c0 3613   ~Pcpw 3823   {csn 3838   U.cuni 4039    e. cmpt 4291   `'ccnv 4906    |` cres 4909   "cima 4910   ` cfv 5483  (class class class)co 6110   ↾t crest 13679   Topctop 16989    Cn ccn 17319    Homeo chmeo 17816   CovMap ccvm 24973
This theorem is referenced by:  cvmcov2  24993  cvmopnlem  24996  cvmfolem  24997  cvmliftmolem2  25000  cvmliftlem15  25016  cvmlift2lem10  25030  cvmlift3lem8  25044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-cvm 24974
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