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Theorem cvmcn 24951
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 2438 . . . 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 2438 . . . 4  |-  U. J  =  U. J
31, 2iscvm 24948 . . 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 448 . 2  |-  ( F  e.  ( C CovMap  J
)  ->  ( C  e.  Top  /\  J  e. 
Top  /\  F  e.  ( C  Cn  J
) ) )
54simp3d 972 1  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   {crab 2711    \ cdif 3319    i^i cin 3321   (/)c0 3630   ~Pcpw 3801   {csn 3816   U.cuni 4017    e. cmpt 4268   `'ccnv 4879    |` cres 4882   "cima 4883   ` cfv 5456  (class class class)co 6083   ↾t crest 13650   Topctop 16960    Cn ccn 17290    Homeo chmeo 17787   CovMap ccvm 24944
This theorem is referenced by:  cvmsss2  24963  cvmseu  24965  cvmopnlem  24967  cvmfolem  24968  cvmliftmolem1  24970  cvmliftmolem2  24971  cvmliftlem6  24979  cvmliftlem7  24980  cvmliftlem8  24981  cvmliftlem9  24982  cvmlift2lem7  24998  cvmlift2lem9  25000  cvmliftphtlem  25006  cvmlift3lem5  25012  cvmlift3lem6  25013  cvmlift3lem9  25016
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-cvm 24945
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