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Theorem cvmtop1 28972
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 3788 . . 3  |-  ( F  e.  ( C CovMap  J
)  ->  -.  ( C CovMap  J )  =  (/) )
2 fncvm 28969 . . . . 5  |- CovMap  Fn  ( Top  X.  Top )
3 fndm 5662 . . . . 5  |-  ( CovMap  Fn  ( Top  X.  Top )  ->  dom CovMap  =  ( Top  X. 
Top ) )
42, 3ax-mp 5 . . . 4  |-  dom CovMap  =  ( Top  X.  Top )
54ndmov 6432 . . 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 457 1  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   (/)c0 3783    X. cxp 4986   dom cdm 4988    Fn wfn 5565  (class class class)co 6270   Topctop 19564   CovMap ccvm 28967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-cvm 28968
This theorem is referenced by:  cvmsf1o  28984  cvmscld  28985  cvmsss2  28986  cvmopnlem  28990  cvmliftmolem1  28993  cvmliftlem8  29004  cvmlift2lem9a  29015  cvmlift2lem9  29023  cvmlift2lem11  29025  cvmlift2lem12  29026  cvmliftphtlem  29029  cvmlift3lem6  29036  cvmlift3lem8  29038  cvmlift3lem9  29039
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