MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  istpsi Unicode version

Theorem istpsi 16682
Description: Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)
Hypotheses
Ref Expression
istpsi.b  |-  ( Base `  K )  =  A
istpsi.j  |-  ( TopOpen `  K )  =  J
istpsi.1  |-  A  = 
U. J
istpsi.2  |-  J  e. 
Top
Assertion
Ref Expression
istpsi  |-  K  e. 
TopSp

Proof of Theorem istpsi
StepHypRef Expression
1 istpsi.2 . 2  |-  J  e. 
Top
2 istpsi.1 . 2  |-  A  = 
U. J
3 istpsi.b . . . 4  |-  ( Base `  K )  =  A
43eqcomi 2287 . . 3  |-  A  =  ( Base `  K
)
5 istpsi.j . . . 4  |-  ( TopOpen `  K )  =  J
65eqcomi 2287 . . 3  |-  J  =  ( TopOpen `  K )
74, 6istps2 16675 . 2  |-  ( K  e.  TopSp 
<->  ( J  e.  Top  /\  A  =  U. J
) )
81, 2, 7mpbir2an 886 1  |-  K  e. 
TopSp
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   U.cuni 3827   ` cfv 5255   Basecbs 13148   TopOpenctopn 13326   Topctop 16631   TopSpctps 16634
This theorem is referenced by:  indistps2  16749
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-top 16636  df-topon 16639  df-topsp 16640
  Copyright terms: Public domain W3C validator