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

Theorem istpsi 19530
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 2395 . . 3  |-  A  =  ( Base `  K
)
5 istpsi.j . . . 4  |-  ( TopOpen `  K )  =  J
65eqcomi 2395 . . 3  |-  J  =  ( TopOpen `  K )
74, 6istps2 19523 . 2  |-  ( K  e.  TopSp 
<->  ( J  e.  Top  /\  A  =  U. J
) )
81, 2, 7mpbir2an 918 1  |-  K  e. 
TopSp
Colors of variables: wff setvar class
Syntax hints:    = wceq 1399    e. wcel 1826   U.cuni 4163   ` cfv 5496   Basecbs 14634   TopOpenctopn 14829   Topctop 19479   TopSpctps 19482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-top 19484  df-topon 19487  df-topsp 19488
This theorem is referenced by:  indistps2  19598
  Copyright terms: Public domain W3C validator