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Theorem cvmtop1 26997
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 3630 . . 3  |-  ( F  e.  ( C CovMap  J
)  ->  -.  ( C CovMap  J )  =  (/) )
2 fncvm 26994 . . . . 5  |- CovMap  Fn  ( Top  X.  Top )
3 fndm 5498 . . . . 5  |-  ( CovMap  Fn  ( Top  X.  Top )  ->  dom CovMap  =  ( Top  X. 
Top ) )
42, 3ax-mp 5 . . . 4  |-  dom CovMap  =  ( Top  X.  Top )
54ndmov 6236 . . 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 456 1  |-  ( F  e.  ( C CovMap  J
)  ->  C  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   (/)c0 3625    X. cxp 4825   dom cdm 4827    Fn wfn 5401  (class class class)co 6080   Topctop 18340   CovMap ccvm 26992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-1st 6566  df-2nd 6567  df-cvm 26993
This theorem is referenced by:  cvmsf1o  27009  cvmscld  27010  cvmsss2  27011  cvmopnlem  27015  cvmliftmolem1  27018  cvmliftlem8  27029  cvmlift2lem9a  27040  cvmlift2lem9  27048  cvmlift2lem11  27050  cvmlift2lem12  27051  cvmliftphtlem  27054  cvmlift3lem6  27061  cvmlift3lem8  27063  cvmlift3lem9  27064
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