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Theorem tgiun 16733
Description: The indexed union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
Assertion
Ref Expression
tgiun  |-  ( ( B  e.  V  /\  A. x  e.  A  C  e.  B )  ->  U_ x  e.  A  C  e.  ( topGen `  B )
)
Distinct variable groups:    x, A    x, B    x, V
Allowed substitution hint:    C( x)

Proof of Theorem tgiun
StepHypRef Expression
1 dfiun3g 4947 . . 3  |-  ( A. x  e.  A  C  e.  B  ->  U_ x  e.  A  C  =  U. ran  ( x  e.  A  |->  C ) )
21adantl 452 . 2  |-  ( ( B  e.  V  /\  A. x  e.  A  C  e.  B )  ->  U_ x  e.  A  C  =  U. ran  ( x  e.  A  |->  C ) )
3 eqid 2296 . . . . 5  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
43fmpt 5697 . . . 4  |-  ( A. x  e.  A  C  e.  B  <->  ( x  e.  A  |->  C ) : A --> B )
5 frn 5411 . . . 4  |-  ( ( x  e.  A  |->  C ) : A --> B  ->  ran  ( x  e.  A  |->  C )  C_  B
)
64, 5sylbi 187 . . 3  |-  ( A. x  e.  A  C  e.  B  ->  ran  (
x  e.  A  |->  C )  C_  B )
7 eltg3i 16715 . . 3  |-  ( ( B  e.  V  /\  ran  ( x  e.  A  |->  C )  C_  B
)  ->  U. ran  (
x  e.  A  |->  C )  e.  ( topGen `  B ) )
86, 7sylan2 460 . 2  |-  ( ( B  e.  V  /\  A. x  e.  A  C  e.  B )  ->  U. ran  ( x  e.  A  |->  C )  e.  (
topGen `  B ) )
92, 8eqeltrd 2370 1  |-  ( ( B  e.  V  /\  A. x  e.  A  C  e.  B )  ->  U_ x  e.  A  C  e.  ( topGen `  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   U.cuni 3843   U_ciun 3921    e. cmpt 4093   ran crn 4706   -->wf 5267   ` cfv 5271   topGenctg 13358
This theorem is referenced by:  txbasval  17317
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-topgen 13360
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