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Theorem iunopn 11853
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 3784 . . 3  |-  ( A. x  e.  A  B  e.  J  ->  U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B } )
21adantl 272 . 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 3124 . . . 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 175 . . 3  |-  ( A. x  e.  A  B  e.  J  ->  { y  |  E. x  e.  A  y  =  B }  C_  J )
5 uniopn 11852 . . 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 281 . 2  |-  ( ( J  e.  Top  /\  A. x  e.  A  B  e.  J )  ->  U. {
y  |  E. x  e.  A  y  =  B }  e.  J
)
72, 6eqeltrd 2171 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 103    = wceq 1296    e. wcel 1445   {cab 2081   A.wral 2370   E.wrex 2371    C_ wss 3013   U.cuni 3675   U_ciun 3752   Topctop 11848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-in 3019  df-ss 3026  df-pw 3451  df-uni 3676  df-iun 3754  df-top 11849
This theorem is referenced by:  tgcn  12059
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