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Theorem cvmcov 24911
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 24907 . . . 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 451 . . 3  |-  ( F  e.  ( C CovMap  J
)  ->  A. x  e.  X  E. k  e.  J  ( x  e.  k  /\  ( S `  k )  =/=  (/) ) )
5 eleq1 2472 . . . . . 6  |-  ( x  =  P  ->  (
x  e.  k  <->  P  e.  k ) )
65anbi1d 686 . . . . 5  |-  ( x  =  P  ->  (
( x  e.  k  /\  ( S `  k )  =/=  (/) )  <->  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) ) )
76rexbidv 2695 . . . 4  |-  ( x  =  P  ->  ( E. k  e.  J  ( x  e.  k  /\  ( S `  k
)  =/=  (/) )  <->  E. k  e.  J  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) ) )
87rspcv 3016 . . 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 456 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  P  e.  X )  ->  E. k  e.  J  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) )
10 nfv 1626 . . . 4  |-  F/ k  P  e.  x
11 nfmpt1 4266 . . . . . . 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 2545 . . . . . 6  |-  F/_ k S
13 nfcv 2548 . . . . . 6  |-  F/_ k
x
1412, 13nffv 5702 . . . . 5  |-  F/_ k
( S `  x
)
15 nfcv 2548 . . . . 5  |-  F/_ k (/)
1614, 15nfne 2666 . . . 4  |-  F/ k ( S `  x
)  =/=  (/)
1710, 16nfan 1842 . . 3  |-  F/ k ( P  e.  x  /\  ( S `  x
)  =/=  (/) )
18 nfv 1626 . . 3  |-  F/ x
( P  e.  k  /\  ( S `  k )  =/=  (/) )
19 eleq2 2473 . . . 4  |-  ( x  =  k  ->  ( P  e.  x  <->  P  e.  k ) )
20 fveq2 5695 . . . . 5  |-  ( x  =  k  ->  ( S `  x )  =  ( S `  k ) )
2120neeq1d 2588 . . . 4  |-  ( x  =  k  ->  (
( S `  x
)  =/=  (/)  <->  ( S `  k )  =/=  (/) ) )
2219, 21anbi12d 692 . . 3  |-  ( x  =  k  ->  (
( P  e.  x  /\  ( S `  x
)  =/=  (/) )  <->  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) ) )
2317, 18, 22cbvrex 2897 . 2  |-  ( E. x  e.  J  ( P  e.  x  /\  ( S `  x )  =/=  (/) )  <->  E. k  e.  J  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) )
249, 23sylibr 204 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   E.wrex 2675   {crab 2678    \ cdif 3285    i^i cin 3287   (/)c0 3596   ~Pcpw 3767   {csn 3782   U.cuni 3983    e. cmpt 4234   `'ccnv 4844    |` cres 4847   "cima 4848   ` cfv 5421  (class class class)co 6048   ↾t crest 13611   Topctop 16921    Cn ccn 17250    Homeo chmeo 17746   CovMap ccvm 24903
This theorem is referenced by:  cvmcov2  24923  cvmopnlem  24926  cvmfolem  24927  cvmliftmolem2  24930  cvmliftlem15  24946  cvmlift2lem10  24960  cvmlift3lem8  24974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-cvm 24904
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