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Theorem iunopn 18642
Description: The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
Assertion
Ref Expression
iunopn  |-  ( ( J  e.  Top  /\  A. x  e.  A  B  e.  J )  ->  U_ x  e.  A  B  e.  J )
Distinct variable groups:    x, A    x, J
Allowed substitution hint:    B( x)

Proof of Theorem iunopn
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfiun2g 4309 . . 3  |-  ( A. x  e.  A  B  e.  J  ->  U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B } )
21adantl 466 . 2  |-  ( ( J  e.  Top  /\  A. x  e.  A  B  e.  J )  ->  U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B } )
3 uniiunlem 3547 . . . 4  |-  ( A. x  e.  A  B  e.  J  ->  ( A. x  e.  A  B  e.  J  <->  { y  |  E. x  e.  A  y  =  B }  C_  J
) )
43ibi 241 . . 3  |-  ( A. x  e.  A  B  e.  J  ->  { y  |  E. x  e.  A  y  =  B }  C_  J )
5 uniopn 18641 . . 3  |-  ( ( J  e.  Top  /\  { y  |  E. x  e.  A  y  =  B }  C_  J )  ->  U. { y  |  E. x  e.  A  y  =  B }  e.  J )
64, 5sylan2 474 . 2  |-  ( ( J  e.  Top  /\  A. x  e.  A  B  e.  J )  ->  U. {
y  |  E. x  e.  A  y  =  B }  e.  J
)
72, 6eqeltrd 2542 1  |-  ( ( J  e.  Top  /\  A. x  e.  A  B  e.  J )  ->  U_ x  e.  A  B  e.  J )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2439   A.wral 2798   E.wrex 2799    C_ wss 3435   U.cuni 4198   U_ciun 4278   Topctop 18629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2803  df-rex 2804  df-v 3078  df-in 3442  df-ss 3449  df-pw 3969  df-uni 4199  df-iun 4280  df-top 18634
This theorem is referenced by:  iincld  18774  tgcn  18987  kgentopon  19242  xkococnlem  19363  qtoptop2  19403  zcld  20521  metnrmlem2  20567  cnambfre  28587  dvtanlem  28588
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