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Theorem tgiun 14050
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 4902 . . 3  |-  ( A. x  e.  A  C  e.  B  ->  U_ x  e.  A  C  =  U. ran  ( x  e.  A  |->  C ) )
21adantl 277 . 2  |-  ( ( B  e.  V  /\  A. x  e.  A  C  e.  B )  ->  U_ x  e.  A  C  =  U. ran  ( x  e.  A  |->  C ) )
3 eqid 2189 . . . 4  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
43rnmptss 5698 . . 3  |-  ( A. x  e.  A  C  e.  B  ->  ran  (
x  e.  A  |->  C )  C_  B )
5 eltg3i 14033 . . 3  |-  ( ( B  e.  V  /\  ran  ( x  e.  A  |->  C )  C_  B
)  ->  U. ran  (
x  e.  A  |->  C )  e.  ( topGen `  B ) )
64, 5sylan2 286 . 2  |-  ( ( B  e.  V  /\  A. x  e.  A  C  e.  B )  ->  U. ran  ( x  e.  A  |->  C )  e.  (
topGen `  B ) )
72, 6eqeltrd 2266 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 104    = wceq 1364    e. wcel 2160   A.wral 2468    C_ wss 3144   U.cuni 3824   U_ciun 3901    |-> cmpt 4079   ran crn 4645   ` cfv 5235   topGenctg 12762
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-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-topgen 12768
This theorem is referenced by:  txbasval  14244
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