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Theorem cvmcn 29546
Description: A covering map is a continuous function. (Contributed by Mario Carneiro, 13-Feb-2015.)
Assertion
Ref Expression
cvmcn  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )

Proof of Theorem cvmcn
Dummy variables  k 
s  u  v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2402 . . . 4  |-  ( 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 ) ) ) ) } )  =  ( 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 eqid 2402 . . . 4  |-  U. J  =  U. J
31, 2iscvm 29543 . . 3  |-  ( F  e.  ( C CovMap  J
)  <->  ( ( C  e.  Top  /\  J  e.  Top  /\  F  e.  ( C  Cn  J
) )  /\  A. x  e.  U. J E. k  e.  J  (
x  e.  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 ) ) ) ) } ) `  k
)  =/=  (/) ) ) )
43simplbi 458 . 2  |-  ( F  e.  ( C CovMap  J
)  ->  ( C  e.  Top  /\  J  e. 
Top  /\  F  e.  ( C  Cn  J
) ) )
54simp3d 1011 1  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2753   E.wrex 2754   {crab 2757    \ cdif 3410    i^i cin 3412   (/)c0 3737   ~Pcpw 3954   {csn 3971   U.cuni 4190    |-> cmpt 4452   `'ccnv 4821    |` cres 4824   "cima 4825   ` cfv 5568  (class class class)co 6277   ↾t crest 15033   Topctop 19684    Cn ccn 20016   Homeochmeo 20544   CovMap ccvm 29539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-cvm 29540
This theorem is referenced by:  cvmsss2  29558  cvmseu  29560  cvmopnlem  29562  cvmfolem  29563  cvmliftmolem1  29565  cvmliftmolem2  29566  cvmliftlem6  29574  cvmliftlem7  29575  cvmliftlem8  29576  cvmliftlem9  29577  cvmlift2lem7  29593  cvmlift2lem9  29595  cvmliftphtlem  29601  cvmlift3lem5  29607  cvmlift3lem6  29608  cvmlift3lem9  29611
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