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Theorem cvmtop2 23807
Description: Reverse closure for a covering map. (Contributed by Mario Carneiro, 13-Feb-2015.)
Assertion
Ref Expression
cvmtop2  |-  ( F  e.  ( C CovMap  J
)  ->  J  e.  Top )

Proof of Theorem cvmtop2
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 ) )
76simprd 449 1  |-  ( F  e.  ( C CovMap  J
)  ->  J  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  cvmsss2  23820  cvmcov2  23821  cvmopnlem  23824  cvmliftlem8  23838  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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