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

Theorem topbas 16710
Description: A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
Assertion
Ref Expression
topbas  |-  ( J  e.  Top  ->  J  e. 
TopBases )

Proof of Theorem topbas
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inopn 16645 . . . . . . . 8  |-  ( ( J  e.  Top  /\  x  e.  J  /\  y  e.  J )  ->  ( x  i^i  y
)  e.  J )
213expb 1152 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( x  e.  J  /\  y  e.  J
) )  ->  (
x  i^i  y )  e.  J )
32adantr 451 . . . . . 6  |-  ( ( ( J  e.  Top  /\  ( x  e.  J  /\  y  e.  J
) )  /\  z  e.  ( x  i^i  y
) )  ->  (
x  i^i  y )  e.  J )
4 simpr 447 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  ( x  e.  J  /\  y  e.  J
) )  /\  z  e.  ( x  i^i  y
) )  ->  z  e.  ( x  i^i  y
) )
5 ssid 3197 . . . . . . 7  |-  ( x  i^i  y )  C_  ( x  i^i  y
)
64, 5jctir 524 . . . . . 6  |-  ( ( ( J  e.  Top  /\  ( x  e.  J  /\  y  e.  J
) )  /\  z  e.  ( x  i^i  y
) )  ->  (
z  e.  ( x  i^i  y )  /\  ( x  i^i  y
)  C_  ( x  i^i  y ) ) )
7 eleq2 2344 . . . . . . . 8  |-  ( w  =  ( x  i^i  y )  ->  (
z  e.  w  <->  z  e.  ( x  i^i  y
) ) )
8 sseq1 3199 . . . . . . . 8  |-  ( w  =  ( x  i^i  y )  ->  (
w  C_  ( x  i^i  y )  <->  ( x  i^i  y )  C_  (
x  i^i  y )
) )
97, 8anbi12d 691 . . . . . . 7  |-  ( w  =  ( x  i^i  y )  ->  (
( z  e.  w  /\  w  C_  ( x  i^i  y ) )  <-> 
( z  e.  ( x  i^i  y )  /\  ( x  i^i  y )  C_  (
x  i^i  y )
) ) )
109rspcev 2884 . . . . . 6  |-  ( ( ( x  i^i  y
)  e.  J  /\  ( z  e.  ( x  i^i  y )  /\  ( x  i^i  y )  C_  (
x  i^i  y )
) )  ->  E. w  e.  J  ( z  e.  w  /\  w  C_  ( x  i^i  y
) ) )
113, 6, 10syl2anc 642 . . . . 5  |-  ( ( ( J  e.  Top  /\  ( x  e.  J  /\  y  e.  J
) )  /\  z  e.  ( x  i^i  y
) )  ->  E. w  e.  J  ( z  e.  w  /\  w  C_  ( x  i^i  y
) ) )
1211exp31 587 . . . 4  |-  ( J  e.  Top  ->  (
( x  e.  J  /\  y  e.  J
)  ->  ( z  e.  ( x  i^i  y
)  ->  E. w  e.  J  ( z  e.  w  /\  w  C_  ( x  i^i  y
) ) ) ) )
1312ralrimdv 2632 . . 3  |-  ( J  e.  Top  ->  (
( x  e.  J  /\  y  e.  J
)  ->  A. z  e.  ( x  i^i  y
) E. w  e.  J  ( z  e.  w  /\  w  C_  ( x  i^i  y
) ) ) )
1413ralrimivv 2634 . 2  |-  ( J  e.  Top  ->  A. x  e.  J  A. y  e.  J  A. z  e.  ( x  i^i  y
) E. w  e.  J  ( z  e.  w  /\  w  C_  ( x  i^i  y
) ) )
15 isbasis2g 16686 . 2  |-  ( J  e.  Top  ->  ( J  e.  TopBases  <->  A. x  e.  J  A. y  e.  J  A. z  e.  (
x  i^i  y ) E. w  e.  J  ( z  e.  w  /\  w  C_  ( x  i^i  y ) ) ) )
1614, 15mpbird 223 1  |-  ( J  e.  Top  ->  J  e. 
TopBases )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   Topctop 16631   TopBasesctb 16635
This theorem is referenced by:  resttop  16891  dis1stc  17225  txtop  17264  onpsstopbas  24869
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627  df-uni 3828  df-top 16636  df-bases 16638
  Copyright terms: Public domain W3C validator