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Theorem cvmcov 28376
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 28372 . . . 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 464 . . 3  |-  ( F  e.  ( C CovMap  J
)  ->  A. x  e.  X  E. k  e.  J  ( x  e.  k  /\  ( S `  k )  =/=  (/) ) )
5 eleq1 2539 . . . . . 6  |-  ( x  =  P  ->  (
x  e.  k  <->  P  e.  k ) )
65anbi1d 704 . . . . 5  |-  ( x  =  P  ->  (
( x  e.  k  /\  ( S `  k )  =/=  (/) )  <->  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) ) )
76rexbidv 2973 . . . 4  |-  ( x  =  P  ->  ( E. k  e.  J  ( x  e.  k  /\  ( S `  k
)  =/=  (/) )  <->  E. k  e.  J  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) ) )
87rspcv 3210 . . 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 469 . 2  |-  ( ( F  e.  ( C CovMap  J )  /\  P  e.  X )  ->  E. k  e.  J  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) )
10 nfv 1683 . . . 4  |-  F/ k  P  e.  x
11 nfmpt1 4536 . . . . . . 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 2627 . . . . . 6  |-  F/_ k S
13 nfcv 2629 . . . . . 6  |-  F/_ k
x
1412, 13nffv 5873 . . . . 5  |-  F/_ k
( S `  x
)
15 nfcv 2629 . . . . 5  |-  F/_ k (/)
1614, 15nfne 2798 . . . 4  |-  F/ k ( S `  x
)  =/=  (/)
1710, 16nfan 1875 . . 3  |-  F/ k ( P  e.  x  /\  ( S `  x
)  =/=  (/) )
18 nfv 1683 . . 3  |-  F/ x
( P  e.  k  /\  ( S `  k )  =/=  (/) )
19 eleq2 2540 . . . 4  |-  ( x  =  k  ->  ( P  e.  x  <->  P  e.  k ) )
20 fveq2 5866 . . . . 5  |-  ( x  =  k  ->  ( S `  x )  =  ( S `  k ) )
2120neeq1d 2744 . . . 4  |-  ( x  =  k  ->  (
( S `  x
)  =/=  (/)  <->  ( S `  k )  =/=  (/) ) )
2219, 21anbi12d 710 . . 3  |-  ( x  =  k  ->  (
( P  e.  x  /\  ( S `  x
)  =/=  (/) )  <->  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) ) )
2317, 18, 22cbvrex 3085 . 2  |-  ( E. x  e.  J  ( P  e.  x  /\  ( S `  x )  =/=  (/) )  <->  E. k  e.  J  ( P  e.  k  /\  ( S `  k )  =/=  (/) ) )
249, 23sylibr 212 1  |-  ( ( F  e.  ( C CovMap  J )  /\  P  e.  X )  ->  E. x  e.  J  ( P  e.  x  /\  ( S `  x )  =/=  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818    \ cdif 3473    i^i cin 3475   (/)c0 3785   ~Pcpw 4010   {csn 4027   U.cuni 4245    |-> cmpt 4505   `'ccnv 4998    |` cres 5001   "cima 5002   ` cfv 5588  (class class class)co 6284   ↾t crest 14676   Topctop 19189    Cn ccn 19519   Homeochmeo 20017   CovMap ccvm 28368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-cvm 28369
This theorem is referenced by:  cvmcov2  28388  cvmopnlem  28391  cvmfolem  28392  cvmliftmolem2  28395  cvmliftlem15  28411  cvmlift2lem10  28425  cvmlift3lem8  28439
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