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Theorem iunopn 13979
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 3933 . . 3  |-  ( A. x  e.  A  B  e.  J  ->  U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B } )
21adantl 277 . 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 3259 . . . 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 176 . . 3  |-  ( A. x  e.  A  B  e.  J  ->  { y  |  E. x  e.  A  y  =  B }  C_  J )
5 uniopn 13978 . . 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 286 . 2  |-  ( ( J  e.  Top  /\  A. x  e.  A  B  e.  J )  ->  U. {
y  |  E. x  e.  A  y  =  B }  e.  J
)
72, 6eqeltrd 2266 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 104    = wceq 1364    e. wcel 2160   {cab 2175   A.wral 2468   E.wrex 2469    C_ wss 3144   U.cuni 3824   U_ciun 3901   Topctop 13974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-sep 4136
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592  df-uni 3825  df-iun 3903  df-top 13975
This theorem is referenced by:  tgcn  14185
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