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Theorem cvmtop1 27292
Description: Reverse closure for a covering map. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
cvmtop1  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )

Proof of Theorem cvmtop1
StepHypRef Expression
1 n0i 3749 . . 3  |-  ( F  e.  ( C CovMap  J
)  ->  -.  ( C CovMap  J )  =  (/) )
2 fncvm 27289 . . . . 5  |- CovMap  Fn  ( Top  X.  Top )
3 fndm 5617 . . . . 5  |-  ( CovMap  Fn  ( Top  X.  Top )  ->  dom CovMap  =  ( Top  X. 
Top ) )
42, 3ax-mp 5 . . . 4  |-  dom CovMap  =  ( Top  X.  Top )
54ndmov 6356 . . 3  |-  ( -.  ( C  e.  Top  /\  J  e.  Top )  ->  ( C CovMap  J )  =  (/) )
61, 5nsyl2 127 . 2  |-  ( F  e.  ( C CovMap  J
)  ->  ( C  e.  Top  /\  J  e. 
Top ) )
76simpld 459 1  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   (/)c0 3744    X. cxp 4945   dom cdm 4947    Fn wfn 5520  (class class class)co 6199   Topctop 18629   CovMap ccvm 27287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1st 6686  df-2nd 6687  df-cvm 27288
This theorem is referenced by:  cvmsf1o  27304  cvmscld  27305  cvmsss2  27306  cvmopnlem  27310  cvmliftmolem1  27313  cvmliftlem8  27324  cvmlift2lem9a  27335  cvmlift2lem9  27343  cvmlift2lem11  27345  cvmlift2lem12  27346  cvmliftphtlem  27349  cvmlift3lem6  27356  cvmlift3lem8  27358  cvmlift3lem9  27359
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