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Theorem cvmcn 27156
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 2443 . . . 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 2443 . . . 4  |-  U. J  =  U. J
31, 2iscvm 27153 . . 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 460 . 2  |-  ( F  e.  ( C CovMap  J
)  ->  ( C  e.  Top  /\  J  e. 
Top  /\  F  e.  ( C  Cn  J
) ) )
54simp3d 1002 1  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   {crab 2724    \ cdif 3330    i^i cin 3332   (/)c0 3642   ~Pcpw 3865   {csn 3882   U.cuni 4096    e. cmpt 4355   `'ccnv 4844    |` cres 4847   "cima 4848   ` cfv 5423  (class class class)co 6096   ↾t crest 14364   Topctop 18503    Cn ccn 18833   Homeochmeo 19331   CovMap ccvm 27149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  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 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-cvm 27150
This theorem is referenced by:  cvmsss2  27168  cvmseu  27170  cvmopnlem  27172  cvmfolem  27173  cvmliftmolem1  27175  cvmliftmolem2  27176  cvmliftlem6  27184  cvmliftlem7  27185  cvmliftlem8  27186  cvmliftlem9  27187  cvmlift2lem7  27203  cvmlift2lem9  27205  cvmliftphtlem  27211  cvmlift3lem5  27217  cvmlift3lem6  27218  cvmlift3lem9  27221
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