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Theorem iunopn 19574
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 4347 . . 3  |-  ( A. x  e.  A  B  e.  J  ->  U_ x  e.  A  B  =  U. { y  |  E. x  e.  A  y  =  B } )
21adantl 464 . 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 3574 . . . 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 19573 . . 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 472 . 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 367    = wceq 1398    e. wcel 1823   {cab 2439   A.wral 2804   E.wrex 2805    C_ wss 3461   U.cuni 4235   U_ciun 4315   Topctop 19561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-v 3108  df-in 3468  df-ss 3475  df-pw 4001  df-uni 4236  df-iun 4317  df-top 19566
This theorem is referenced by:  iincld  19707  tgcn  19920  kgentopon  20205  xkococnlem  20326  qtoptop2  20366  zcld  21484  metnrmlem2  21530  cnambfre  30303  dvtanlem  30304
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