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Theorem inopn 16974
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 16970 . . . . 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 234 . . . 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 451 . . 3  |-  ( J  e.  Top  ->  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J
)
4 ineq1 3537 . . . . 5  |-  ( x  =  A  ->  (
x  i^i  y )  =  ( A  i^i  y ) )
54eleq1d 2504 . . . 4  |-  ( x  =  A  ->  (
( x  i^i  y
)  e.  J  <->  ( A  i^i  y )  e.  J
) )
6 ineq2 3538 . . . . 5  |-  ( y  =  B  ->  ( A  i^i  y )  =  ( A  i^i  B
) )
76eleq1d 2504 . . . 4  |-  ( y  =  B  ->  (
( A  i^i  y
)  e.  J  <->  ( A  i^i  B )  e.  J
) )
85, 7rspc2v 3060 . . 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 29 . 2  |-  ( J  e.  Top  ->  (
( A  e.  J  /\  B  e.  J
)  ->  ( A  i^i  B )  e.  J
) )
1093impib 1152 1  |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  ( A  i^i  B
)  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937   A.wal 1550    = wceq 1653    e. wcel 1726   A.wral 2707    i^i cin 3321    C_ wss 3322   U.cuni 4017   Topctop 16960
This theorem is referenced by:  fitop  16975  tgclb  17037  topbas  17039  difopn  17100  uncld  17107  ntrin  17127  toponmre  17159  innei  17191  restopnb  17241  ordtopn3  17262  cnprest  17355  islly2  17549  kgentopon  17572  llycmpkgen2  17584  ptbasin  17611  txcnp  17654  txcnmpt  17658  qtoptop2  17733  opnfbas  17876  hauspwpwf1  18021  mopnin  18529  reconnlem2  18860  lmxrge0  24339  cvmsss2  24963  cvmcov2  24964
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-v 2960  df-in 3329  df-ss 3336  df-pw 3803  df-top 16965
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