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Theorem cvmcn 23793
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 2283 . . . 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 2283 . . . 4  |-  U. J  =  U. J
31, 2iscvm 23790 . . 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 446 . 2  |-  ( F  e.  ( C CovMap  J
)  ->  ( C  e.  Top  /\  J  e. 
Top  /\  F  e.  ( C  Cn  J
) ) )
54simp3d 969 1  |-  ( F  e.  ( C CovMap  J
)  ->  F  e.  ( C  Cn  J
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547    \ cdif 3149    i^i cin 3151   (/)c0 3455   ~Pcpw 3625   {csn 3640   U.cuni 3827    e. cmpt 4077   `'ccnv 4688    |` cres 4691   "cima 4692   ` cfv 5255  (class class class)co 5858   ↾t crest 13325   Topctop 16631    Cn ccn 16954    Homeo chmeo 17444   CovMap ccvm 23786
This theorem is referenced by:  cvmsss2  23805  cvmseu  23807  cvmopnlem  23809  cvmfolem  23810  cvmliftmolem1  23812  cvmliftmolem2  23813  cvmliftlem6  23821  cvmliftlem7  23822  cvmliftlem8  23823  cvmliftlem9  23824  cvmlift2lem7  23840  cvmlift2lem9  23842  cvmliftphtlem  23848  cvmlift3lem5  23854  cvmlift3lem6  23855  cvmlift3lem9  23858
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-cvm 23787
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