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Theorem tgiun 19247
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 5253 . . 3  |-  ( A. x  e.  A  C  e.  B  ->  U_ x  e.  A  C  =  U. ran  ( x  e.  A  |->  C ) )
21adantl 466 . 2  |-  ( ( B  e.  V  /\  A. x  e.  A  C  e.  B )  ->  U_ x  e.  A  C  =  U. ran  ( x  e.  A  |->  C ) )
3 eqid 2467 . . . . 5  |-  ( x  e.  A  |->  C )  =  ( x  e.  A  |->  C )
43fmpt 6040 . . . 4  |-  ( A. x  e.  A  C  e.  B  <->  ( x  e.  A  |->  C ) : A --> B )
5 frn 5735 . . . 4  |-  ( ( x  e.  A  |->  C ) : A --> B  ->  ran  ( x  e.  A  |->  C )  C_  B
)
64, 5sylbi 195 . . 3  |-  ( A. x  e.  A  C  e.  B  ->  ran  (
x  e.  A  |->  C )  C_  B )
7 eltg3i 19229 . . 3  |-  ( ( B  e.  V  /\  ran  ( x  e.  A  |->  C )  C_  B
)  ->  U. ran  (
x  e.  A  |->  C )  e.  ( topGen `  B ) )
86, 7sylan2 474 . 2  |-  ( ( B  e.  V  /\  A. x  e.  A  C  e.  B )  ->  U. ran  ( x  e.  A  |->  C )  e.  (
topGen `  B ) )
92, 8eqeltrd 2555 1  |-  ( ( B  e.  V  /\  A. x  e.  A  C  e.  B )  ->  U_ x  e.  A  C  e.  ( topGen `  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814    C_ wss 3476   U.cuni 4245   U_ciun 4325    |-> cmpt 4505   ran crn 5000   -->wf 5582   ` cfv 5586   topGenctg 14689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-topgen 14695
This theorem is referenced by:  txbasval  19842
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