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Theorem istps2 16994
Description: Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.)
Hypotheses
Ref Expression
istps.a  |-  A  =  ( Base `  K
)
istps.j  |-  J  =  ( TopOpen `  K )
Assertion
Ref Expression
istps2  |-  ( K  e.  TopSp 
<->  ( J  e.  Top  /\  A  =  U. J
) )

Proof of Theorem istps2
StepHypRef Expression
1 istps.a . . 3  |-  A  =  ( Base `  K
)
2 istps.j . . 3  |-  J  =  ( TopOpen `  K )
31, 2istps 16993 . 2  |-  ( K  e.  TopSp 
<->  J  e.  (TopOn `  A ) )
4 istopon 16982 . 2  |-  ( J  e.  (TopOn `  A
)  <->  ( J  e. 
Top  /\  A  =  U. J ) )
53, 4bitri 241 1  |-  ( K  e.  TopSp 
<->  ( J  e.  Top  /\  A  =  U. J
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   U.cuni 4007   ` cfv 5446   Basecbs 13461   TopOpenctopn 13641   Topctop 16950  TopOnctopon 16951   TopSpctps 16953
This theorem is referenced by:  tpsuni  16995  tpstop  16996  istpsi  17001
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-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  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-iota 5410  df-fun 5448  df-fv 5454  df-top 16955  df-topon 16958  df-topsp 16959
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