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Theorem iunopn 16644
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 3935 . . 3  |-  ( A. x  e.  A  B  e.  J  ->  U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B } )
21adantl 452 . 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 3260 . . . 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 232 . . 3  |-  ( A. x  e.  A  B  e.  J  ->  { y  |  E. x  e.  A  y  =  B }  C_  J )
5 uniopn 16643 . . 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 460 . 2  |-  ( ( J  e.  Top  /\  A. x  e.  A  B  e.  J )  ->  U. {
y  |  E. x  e.  A  y  =  B }  e.  J
)
72, 6eqeltrd 2357 1  |-  ( ( J  e.  Top  /\  A. x  e.  A  B  e.  J )  ->  U_ x  e.  A  B  e.  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544    C_ wss 3152   U.cuni 3827   U_ciun 3905   Topctop 16631
This theorem is referenced by:  iincld  16776  tgcn  16982  kgentopon  17233  xkococnlem  17353  qtoptop2  17390  zcld  18319  metnrmlem2  18364
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627  df-uni 3828  df-iun 3907  df-top 16636
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