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Theorem 2ndci 17190
Description: A countable basis generates a second-countable topology. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
2ndci  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  ( topGen `  B )  e.  2ndc )

Proof of Theorem 2ndci
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  B  e.  TopBases )
2 simpr 447 . . 3  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  B  ~<_  om )
3 eqidd 2297 . . 3  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  ( topGen `  B )  =  (
topGen `  B ) )
4 breq1 4042 . . . . 5  |-  ( x  =  B  ->  (
x  ~<_  om  <->  B  ~<_  om )
)
5 fveq2 5541 . . . . . 6  |-  ( x  =  B  ->  ( topGen `
 x )  =  ( topGen `  B )
)
65eqeq1d 2304 . . . . 5  |-  ( x  =  B  ->  (
( topGen `  x )  =  ( topGen `  B
)  <->  ( topGen `  B
)  =  ( topGen `  B ) ) )
74, 6anbi12d 691 . . . 4  |-  ( x  =  B  ->  (
( x  ~<_  om  /\  ( topGen `  x )  =  ( topGen `  B
) )  <->  ( B  ~<_  om  /\  ( topGen `  B
)  =  ( topGen `  B ) ) ) )
87rspcev 2897 . . 3  |-  ( ( B  e.  TopBases  /\  ( B  ~<_  om  /\  ( topGen `
 B )  =  ( topGen `  B )
) )  ->  E. x  e. 
TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  ( topGen `  B
) ) )
91, 2, 3, 8syl12anc 1180 . 2  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  E. x  e. 
TopBases  ( x  ~<_  om  /\  ( topGen `  x )  =  ( topGen `  B
) ) )
10 is2ndc 17188 . 2  |-  ( (
topGen `  B )  e. 
2ndc 
<->  E. x  e.  TopBases  ( x  ~<_  om  /\  ( topGen `
 x )  =  ( topGen `  B )
) )
119, 10sylibr 203 1  |-  ( ( B  e.  TopBases  /\  B  ~<_  om )  ->  ( topGen `  B )  e.  2ndc )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039   omcom 4672   ` cfv 5271    ~<_ cdom 6877   topGenctg 13358   TopBasesctb 16651   2ndcc2ndc 17180
This theorem is referenced by:  2ndcrest  17196  2ndcomap  17200  dis2ndc  17202  dis1stc  17241  tx2ndc  17361  met2ndci  18084  re2ndc  18323
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
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-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-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-2ndc 17182
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