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Theorem cvmtop1 23806
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 3473 . . 3  |-  ( F  e.  ( C CovMap  J
)  ->  -.  ( C CovMap  J )  =  (/) )
2 fncvm 23803 . . . . 5  |- CovMap  Fn  ( Top  X.  Top )
3 fndm 5359 . . . . 5  |-  ( CovMap  Fn  ( Top  X.  Top )  ->  dom CovMap  =  ( Top  X. 
Top ) )
42, 3ax-mp 8 . . . 4  |-  dom CovMap  =  ( Top  X.  Top )
54ndmov 6020 . . 3  |-  ( -.  ( C  e.  Top  /\  J  e.  Top )  ->  ( C CovMap  J )  =  (/) )
61, 5nsyl2 119 . 2  |-  ( F  e.  ( C CovMap  J
)  ->  ( C  e.  Top  /\  J  e. 
Top ) )
76simpld 445 1  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   (/)c0 3468    X. cxp 4703   dom cdm 4705    Fn wfn 5266  (class class class)co 5874   Topctop 16647   CovMap ccvm 23801
This theorem is referenced by:  cvmsf1o  23818  cvmscld  23819  cvmsss2  23820  cvmopnlem  23824  cvmliftmolem1  23827  cvmliftlem8  23838  cvmlift2lem9a  23849  cvmlift2lem9  23857  cvmlift2lem11  23859  cvmlift2lem12  23860  cvmliftphtlem  23863  cvmlift3lem6  23870  cvmlift3lem8  23872  cvmlift3lem9  23873
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-cvm 23802
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