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Theorem tgiun 17045
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 5123 . . 3  |-  ( A. x  e.  A  C  e.  B  ->  U_ x  e.  A  C  =  U. ran  ( x  e.  A  |->  C ) )
21adantl 454 . 2  |-  ( ( B  e.  V  /\  A. x  e.  A  C  e.  B )  ->  U_ x  e.  A  C  =  U. ran  ( x  e.  A  |->  C ) )
3 eqid 2437 . . . . 5  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
43fmpt 5891 . . . 4  |-  ( A. x  e.  A  C  e.  B  <->  ( x  e.  A  |->  C ) : A --> B )
5 frn 5598 . . . 4  |-  ( ( x  e.  A  |->  C ) : A --> B  ->  ran  ( x  e.  A  |->  C )  C_  B
)
64, 5sylbi 189 . . 3  |-  ( A. x  e.  A  C  e.  B  ->  ran  (
x  e.  A  |->  C )  C_  B )
7 eltg3i 17027 . . 3  |-  ( ( B  e.  V  /\  ran  ( x  e.  A  |->  C )  C_  B
)  ->  U. ran  (
x  e.  A  |->  C )  e.  ( topGen `  B ) )
86, 7sylan2 462 . 2  |-  ( ( B  e.  V  /\  A. x  e.  A  C  e.  B )  ->  U. ran  ( x  e.  A  |->  C )  e.  (
topGen `  B ) )
92, 8eqeltrd 2511 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 360    = wceq 1653    e. wcel 1726   A.wral 2706    C_ wss 3321   U.cuni 4016   U_ciun 4094    e. cmpt 4267   ran crn 4880   -->wf 5451   ` cfv 5455   topGenctg 13666
This theorem is referenced by:  txbasval  17639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-topgen 13668
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