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Theorem topbas 18536
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 18471 . . . . . . . 8  |-  ( ( J  e.  Top  /\  x  e.  J  /\  y  e.  J )  ->  ( x  i^i  y
)  e.  J )
213expb 1183 . . . . . . 7  |-  ( ( J  e.  Top  /\  ( x  e.  J  /\  y  e.  J
) )  ->  (
x  i^i  y )  e.  J )
32adantr 462 . . . . . 6  |-  ( ( ( J  e.  Top  /\  ( x  e.  J  /\  y  e.  J
) )  /\  z  e.  ( x  i^i  y
) )  ->  (
x  i^i  y )  e.  J )
4 simpr 458 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  ( x  e.  J  /\  y  e.  J
) )  /\  z  e.  ( x  i^i  y
) )  ->  z  e.  ( x  i^i  y
) )
5 ssid 3372 . . . . . . 7  |-  ( x  i^i  y )  C_  ( x  i^i  y
)
64, 5jctir 535 . . . . . 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 2502 . . . . . . . 8  |-  ( w  =  ( x  i^i  y )  ->  (
z  e.  w  <->  z  e.  ( x  i^i  y
) ) )
8 sseq1 3374 . . . . . . . 8  |-  ( w  =  ( x  i^i  y )  ->  (
w  C_  ( x  i^i  y )  <->  ( x  i^i  y )  C_  (
x  i^i  y )
) )
97, 8anbi12d 705 . . . . . . 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 3070 . . . . . 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 656 . . . . 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 601 . . . 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 2803 . . 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 2805 . 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 18512 . 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 232 1  |-  ( J  e.  Top  ->  J  e. 
TopBases )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   E.wrex 2714    i^i cin 3324    C_ wss 3325   Topctop 18457   TopBasesctb 18461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rex 2719  df-v 2972  df-in 3332  df-ss 3339  df-pw 3859  df-uni 4089  df-top 18462  df-bases 18464
This theorem is referenced by:  resttop  18723  dis1stc  19062  txtop  19101  onpsstopbas  28206
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