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Theorem inopn 19698
Description: The intersection of two open sets of a topology is also an open set. (Contributed by NM, 17-Jul-2006.)
Assertion
Ref Expression
inopn  |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  ( A  i^i  B
)  e.  J )

Proof of Theorem inopn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istopg 19694 . . . . 5  |-  ( J  e.  Top  ->  ( J  e.  Top  <->  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J
) ) )
21ibi 241 . . . 4  |-  ( J  e.  Top  ->  ( A. x ( x  C_  J  ->  U. x  e.  J
)  /\  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J
) )
32simprd 461 . . 3  |-  ( J  e.  Top  ->  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J
)
4 ineq1 3633 . . . . 5  |-  ( x  =  A  ->  (
x  i^i  y )  =  ( A  i^i  y ) )
54eleq1d 2471 . . . 4  |-  ( x  =  A  ->  (
( x  i^i  y
)  e.  J  <->  ( A  i^i  y )  e.  J
) )
6 ineq2 3634 . . . . 5  |-  ( y  =  B  ->  ( A  i^i  y )  =  ( A  i^i  B
) )
76eleq1d 2471 . . . 4  |-  ( y  =  B  ->  (
( A  i^i  y
)  e.  J  <->  ( A  i^i  B )  e.  J
) )
85, 7rspc2v 3168 . . 3  |-  ( ( A  e.  J  /\  B  e.  J )  ->  ( A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J  ->  ( A  i^i  B
)  e.  J ) )
93, 8syl5com 28 . 2  |-  ( J  e.  Top  ->  (
( A  e.  J  /\  B  e.  J
)  ->  ( A  i^i  B )  e.  J
) )
1093impib 1195 1  |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  ( A  i^i  B
)  e.  J )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974   A.wal 1403    = wceq 1405    e. wcel 1842   A.wral 2753    i^i cin 3412    C_ wss 3413   U.cuni 4190   Topctop 19684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-v 3060  df-in 3420  df-ss 3427  df-pw 3956  df-top 19689
This theorem is referenced by:  fitop  19699  tgclb  19762  topbas  19764  difopn  19825  uncld  19832  ntrin  19852  toponmre  19885  innei  19917  restopnb  19967  ordtopn3  19988  cnprest  20081  islly2  20275  kgentopon  20329  llycmpkgen2  20341  ptbasin  20368  txcnp  20411  txcnmpt  20415  qtoptop2  20490  opnfbas  20633  hauspwpwf1  20778  mopnin  21290  reconnlem2  21622  lmxrge0  28373  cvmsss2  29558  cvmcov2  29559  icccncfext  37039
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