Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvmcn Unicode version

Theorem cvmcn 24902
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 2404 . . . 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 2404 . . . 4  |-  U. J  =  U. J
31, 2iscvm 24899 . . 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 447 . 2  |-  ( F  e.  ( C CovMap  J
)  ->  ( C  e.  Top  /\  J  e. 
Top  /\  F  e.  ( C  Cn  J
) ) )
54simp3d 971 1  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670    \ cdif 3277    i^i cin 3279   (/)c0 3588   ~Pcpw 3759   {csn 3774   U.cuni 3975    e. cmpt 4226   `'ccnv 4836    |` cres 4839   "cima 4840   ` cfv 5413  (class class class)co 6040   ↾t crest 13603   Topctop 16913    Cn ccn 17242    Homeo chmeo 17738   CovMap ccvm 24895
This theorem is referenced by:  cvmsss2  24914  cvmseu  24916  cvmopnlem  24918  cvmfolem  24919  cvmliftmolem1  24921  cvmliftmolem2  24922  cvmliftlem6  24930  cvmliftlem7  24931  cvmliftlem8  24932  cvmliftlem9  24933  cvmlift2lem7  24949  cvmlift2lem9  24951  cvmliftphtlem  24957  cvmlift3lem5  24963  cvmlift3lem6  24964  cvmlift3lem9  24967
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-cvm 24896
  Copyright terms: Public domain W3C validator