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

Theorem topbas 17037
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 16972 . . . . . . . 8  |-  ( ( J  e.  Top  /\  x  e.  J  /\  y  e.  J )  ->  ( x  i^i  y
)  e.  J )
213expb 1154 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( x  e.  J  /\  y  e.  J
) )  ->  (
x  i^i  y )  e.  J )
32adantr 452 . . . . . 6  |-  ( ( ( J  e.  Top  /\  ( x  e.  J  /\  y  e.  J
) )  /\  z  e.  ( x  i^i  y
) )  ->  (
x  i^i  y )  e.  J )
4 simpr 448 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  ( x  e.  J  /\  y  e.  J
) )  /\  z  e.  ( x  i^i  y
) )  ->  z  e.  ( x  i^i  y
) )
5 ssid 3367 . . . . . . 7  |-  ( x  i^i  y )  C_  ( x  i^i  y
)
64, 5jctir 525 . . . . . 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 2497 . . . . . . . 8  |-  ( w  =  ( x  i^i  y )  ->  (
z  e.  w  <->  z  e.  ( x  i^i  y
) ) )
8 sseq1 3369 . . . . . . . 8  |-  ( w  =  ( x  i^i  y )  ->  (
w  C_  ( x  i^i  y )  <->  ( x  i^i  y )  C_  (
x  i^i  y )
) )
97, 8anbi12d 692 . . . . . . 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 3052 . . . . . 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 643 . . . . 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 588 . . . 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 2795 . . 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 2797 . 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 17013 . 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 224 1  |-  ( J  e.  Top  ->  J  e. 
TopBases )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    i^i cin 3319    C_ wss 3320   Topctop 16958   TopBasesctb 16962
This theorem is referenced by:  resttop  17224  dis1stc  17562  txtop  17601  onpsstopbas  26180
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330
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-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-in 3327  df-ss 3334  df-pw 3801  df-uni 4016  df-top 16963  df-bases 16965
  Copyright terms: Public domain W3C validator