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Theorem cvmtop1 28345
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 3790 . . 3  |-  ( F  e.  ( C CovMap  J
)  ->  -.  ( C CovMap  J )  =  (/) )
2 fncvm 28342 . . . . 5  |- CovMap  Fn  ( Top  X.  Top )
3 fndm 5678 . . . . 5  |-  ( CovMap  Fn  ( Top  X.  Top )  ->  dom CovMap  =  ( Top  X. 
Top ) )
42, 3ax-mp 5 . . . 4  |-  dom CovMap  =  ( Top  X.  Top )
54ndmov 6441 . . 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 1379    e. wcel 1767   (/)c0 3785    X. cxp 4997   dom cdm 4999    Fn wfn 5581  (class class class)co 6282   Topctop 19161   CovMap ccvm 28340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-cvm 28341
This theorem is referenced by:  cvmsf1o  28357  cvmscld  28358  cvmsss2  28359  cvmopnlem  28363  cvmliftmolem1  28366  cvmliftlem8  28377  cvmlift2lem9a  28388  cvmlift2lem9  28396  cvmlift2lem11  28398  cvmlift2lem12  28399  cvmliftphtlem  28402  cvmlift3lem6  28409  cvmlift3lem8  28411  cvmlift3lem9  28412
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