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Theorem inopn 12097
Description: The intersection of two open sets of a topology is 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 12093 . . . . 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 175 . . . 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 113 . . 3  |-  ( J  e.  Top  ->  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J
)
4 ineq1 3240 . . . . 5  |-  ( x  =  A  ->  (
x  i^i  y )  =  ( A  i^i  y ) )
54eleq1d 2186 . . . 4  |-  ( x  =  A  ->  (
( x  i^i  y
)  e.  J  <->  ( A  i^i  y )  e.  J
) )
6 ineq2 3241 . . . . 5  |-  ( y  =  B  ->  ( A  i^i  y )  =  ( A  i^i  B
) )
76eleq1d 2186 . . . 4  |-  ( y  =  B  ->  (
( A  i^i  y
)  e.  J  <->  ( A  i^i  B )  e.  J
) )
85, 7rspc2v 2776 . . 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 1164 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 103    /\ w3a 947   A.wal 1314    = wceq 1316    e. wcel 1465   A.wral 2393    i^i cin 3040    C_ wss 3041   U.cuni 3706   Topctop 12091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482  df-top 12092
This theorem is referenced by:  tgclb  12161  topbas  12163  difopn  12204  uncld  12209  ntrin  12220  innei  12259  restopnb  12277  cnptoprest  12335  txcnp  12367  txcnmpt  12369  mopnin  12583
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