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Theorem topfne 31015
Description: Fineness for covers corresponds precisely with fineness for topologies. (Contributed by Jeff Hankins, 29-Sep-2009.)
Hypotheses
Ref Expression
topfne.1  |-  X  = 
U. J
topfne.2  |-  Y  = 
U. K
Assertion
Ref Expression
topfne  |-  ( ( K  e.  Top  /\  X  =  Y )  ->  ( J  C_  K  <->  J Fne K ) )

Proof of Theorem topfne
StepHypRef Expression
1 tgtop 19987 . . . 4  |-  ( K  e.  Top  ->  ( topGen `
 K )  =  K )
21sseq2d 3492 . . 3  |-  ( K  e.  Top  ->  ( J  C_  ( topGen `  K
)  <->  J  C_  K ) )
32bicomd 204 . 2  |-  ( K  e.  Top  ->  ( J  C_  K  <->  J  C_  ( topGen `
 K ) ) )
4 topfne.1 . . . 4  |-  X  = 
U. J
5 topfne.2 . . . 4  |-  Y  = 
U. K
64, 5isfne4 31001 . . 3  |-  ( J Fne K  <->  ( X  =  Y  /\  J  C_  ( topGen `  K )
) )
76baibr 912 . 2  |-  ( X  =  Y  ->  ( J  C_  ( topGen `  K
)  <->  J Fne K ) )
83, 7sylan9bb 704 1  |-  ( ( K  e.  Top  /\  X  =  Y )  ->  ( J  C_  K  <->  J Fne K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872    C_ wss 3436   U.cuni 4219   class class class wbr 4423   ` cfv 5601   topGenctg 15335   Topctop 19915   Fnecfne 30997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-topgen 15341  df-top 19919  df-fne 30998
This theorem is referenced by: (None)
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