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Theorem cvmtop2 24940
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 3625 . . 3  |-  ( F  e.  ( C CovMap  J
)  ->  -.  ( C CovMap  J )  =  (/) )
2 fncvm 24936 . . . . 5  |- CovMap  Fn  ( Top  X.  Top )
3 fndm 5536 . . . . 5  |-  ( CovMap  Fn  ( Top  X.  Top )  ->  dom CovMap  =  ( Top  X. 
Top ) )
42, 3ax-mp 8 . . . 4  |-  dom CovMap  =  ( Top  X.  Top )
54ndmov 6223 . . 3  |-  ( -.  ( C  e.  Top  /\  J  e.  Top )  ->  ( C CovMap  J )  =  (/) )
61, 5nsyl2 121 . 2  |-  ( F  e.  ( C CovMap  J
)  ->  ( C  e.  Top  /\  J  e. 
Top ) )
76simprd 450 1  |-  ( F  e.  ( C CovMap  J
)  ->  J  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   (/)c0 3620    X. cxp 4868   dom cdm 4870    Fn wfn 5441  (class class class)co 6073   Topctop 16950   CovMap ccvm 24934
This theorem is referenced by:  cvmsf1o  24951  cvmsss2  24953  cvmcov2  24954  cvmopnlem  24957  cvmliftlem8  24971  cvmlift3lem9  25006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-cvm 24935
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